I live in Nova Scotia, and I have been fighting with this weed for at least 7 years. It is in my flower garden. I have gone so far as to try to follow each of the white roots though the soil and remove them and still it comes back. It seems the more you break the roots, the more plants it grows. I have tried Round-Up and other such killers and none have worked but its roots are through my Hostas and this makes it hard to get it all. This year I was down with an operation and it got ahead of me and I need help. I would love to get rid of it once and for all. Thanks for any help you can offer.
If the methods described above aren't any you want to do, there is something else to try, but again, it's not a quick, easy solution - it requires determination and persistence. This plant here in the UK has the common name of ground elder, and it is indeed a pernicious weed, but it is possible to get rid of it for good, other than the odd seedling that may attempt to regrow occasionally.
The method which I found successful was to spray or water on glyphosate when the plant was growing strongly, usually in late spring/early summer, wait about 10-14 days, respray, wait a week, and then dig. It may look as if the ground elder topgrowth is dead, but what the glyphosate seems to do is kill off the fine, hairlike roots which are difficult to see and dig out, leaving behind the more visible root system, which can then be carefully and rigorously extracted.
Your problem is that the roots are in your hostas - that means you need to get the roots of the elder out, so treat the growth you can see amongst the hostas with a thick version of glyphosate (Round up makes a gel version). Then you need to dig up your hostas, preferably late summer, divide if they need it, which hopefully they do, and attempt to remove all and any ground elder root hidden in the rootball. Nova Scotia has various USDA zones - if yours is zone 8, I'd risk potting the hostas till you can see if there's any regrowth of the elder among them, but if your zone is 6, then it might not be possible to keep the hostas in pots over winter because of freeze risk. Otherwise, you can do it in spring, which is the other time that hostas can be divided, but the ground elder may not be in full growth at that time.
You can try this method in the ground now only if the ground elder is still growing strongly, when glyphosate works best on plants - otherwise you will need to wait till next year, but I'd certainly make a start by dealing with the hostas this year.
This plant is commonly known as goutweed (Aegopodium podagraria). Here in the Netherlands it is also called "tuinmansverdriet" (Gardener's Sorrow), and for good reason. I too have this stuff in my garden — a present from the previous home owner — and it's very hard to eliminate.
In my father's garden we got rid of it fairly effectively by excavating the top 30 cm of the soil with a pitchfork (instead of using a spade, in order to prevent cutting up the roots too much) and sifting it all; but even then the weed came back here and there, because even small parts of the root continued to grow.
Currently I'm trying to root it out area by area in my own garden, keeping close track of the locations where I've dealt with it previously, trying to dig out any new growth that surfaces, and (as a minimum) picking off the leaves as soon as I can where uprooting it is impossible due to the presence of other plants nearby.
Once the soil is loose, it is also easier to remove new plants that grow from leftover root fragments.
It seems as if the plant can be at least be contained in this manner, but I doubt if I will be able to completely get rid of it like this, especially around shrubs. You can eliminate it by constantly picking off all its leaves (almost every plant will die eventually if you do this), but it's a really tough plant and this method will take a long time.
So if you are unable or unwilling to get down on your knees once or twice a week, you might need to use a herbicide, as has been suggested in the other answers. If you do this, glyphosate (Roundup) should work, but it is likely that it will grow back at least once after being treated with Roundup.
Covering the soil, as @jmusser suggested, might also work, but my understanding is that you need to leave it covered for at least two years for the plant to die off completely. This was something I was unwilling to do, so I haven't tried this control method.
As long as you are stuck with this weed, try to make the best of it. The plant isn't too unattractive in appearance, and has pretty white flowers. It is also edible, and is actually quite tasty. It can substitute for spinach when cooked and parsley when raw, and it contains lots of minerals and vitamins — though I think it would be wise not to eat it if you have used herbicide on it.
Another tip: if you dig out this plant, don't try to compost it. That would be a very good way to end up with a garden full of it.
As I said, I had the same problem as the questioner, and although digging out small patches seems to work, I couldn't get rid of all of it with all the bushes (plus a hosta) still remaining in my garden. Since I certainly don't want to use Roundup, I did what my dad had done before: digging up the whole border, I removed the plants and bushes I had there and threw them all out.
Fortunately, the infested area measured only about 10 m2. I decided to sift all the soil, though I could just have discarded it and replaced it with new soil instead. It would not have been that expensive to replace, but removing soil is a tough job in itself, and I felt this would also be wasteful. So I sifted it using a piece of netting (or you can buy a soil sieve), removing at least the largest of the roots and other parts of the weed. I actually felt pretty zen after two days of sifting. :D
I put back the by-now-finely-sifted soil, and added compost in order to maintain the loose consistency of the soil. I knew I had not removed all the roots. Small ones passed through the sieve, and sifting the soil again through a finer mesh would have taken too much time and still wouldn't have solved the problem completely. So I knew the remaining fragments would grow back; but having an empty garden full of loosened soil will allow me to easily dig out any new shoots that emerge.
It's easy to recognise the plant. If it emerges from a seed, its first leaves are round seed leaves, but a piece of existing root that grows into a new plant exhibits the typical jagged-edged leaves right from the start.
So now I am able to spot these plants easily, and to dig them out as soon as I do. And that's how, less than two weeks after sifting the soil, I was able to dig up the plant in the picture. Only its first two centimeters consist of pre-existing root. The white stalk is all new (underground) growth, with only the green leaf showing above the surface of the soil. This is proof of the tremendous growing power of this weed, but if I just keep digging out any new sprouts for a couple of months it should be gone completely. Only then will I put in new plants. If I put them in now, the goutweed will certainly come back in places where I can't properly get at it to root it out.
It sounds like a lot of work, but it's not that bad, and not that arduous. You can do the sifting sitting on a chair or stool, and you can take as much time as you like, only doing small bits at a time. For me, the hard part was digging out the existing plants, but you may be able to get some help with that task.
That is Bishop's weed or Goutweed, Aegopodium podagraria, an aggressively growing invasive perennial plant which spreads quickly by underground rhizomes. It is difficult to control, as it is able to quickly regrow from broken pieces of rhizomes left in the soil, as well as root fragments. I have found that glufosinate based herbicides bring better long-term results than glyphosate (Roundup). Neither of these is extremely potent on this weed.
I use herbicides to weaken the plant, and when they begin to shrivel I cut them off at ground level. After that, I take some time to go out with a pair of scissors on my way to work, and snip off any new growth before it gets a chance to feed the roots. The plants will eventually die.
If you have a wide space with nothing else planted, you can cover the area with 4-6mil black plastic sheeting. Leave this on the area for at least two months of hot weather. After that, turn over a small portion, and dig around a little. If there are any roots which are still alive, replace the cover. They will eventually die. If you feel confident that the roots are dead, you can remove the cover, and resume your gardening.
Also, I'd recommend that you dig up infested hostas after they die back in the fall, and completely clean out their crowns, until you are sure you have left no goutweed roots. This will be easier when the soil is not very wet.
If you do not care about anything but destroying the weeds, you can mow over your entire bed(s), and spray with glufosinate. Every time you mow your lawn, go over the bed, and every other week spray with glufosinate until there is nothing green in the bed. After that, only spray when the bed looks like it's greening up. Eventually, the weeds will die.
All methods of control will take time, probably more than a year. Also, you will want to watch for seedlings during the entire process. They are easy enough to control when they are that small.
If you can't beat them, eat them! That's my motto! I've also had no luck eradicating that exact same weed, we call it Kirskål in Sweden (Aegopodium podagraria or Bishop's Weed in English). Each individual root, even the tiniest bit, will continue to grow. But they're edible - the leaves are best eaten when they're small and their leaves just beginning to unfurl. They're supposed to be nutritious, although not the best tasting. I've found that sauteing them in Asian recipes (bold flavors) tastes best!
Eta: The leaves shouldn't be eaten after the plant flowers.
You could try laying down sheets of cardboard , then thick layer of wood chips for mulch. After a year, likely all that's under cardboard should be dead works well on grass you don't want. Best type cardboard I've found is the large sheets , about 3 ft square, found at stores like Costco -- used to separate boxes of canned goods, or packaged items like toilet paper. It's flexible, easy to tear or cut to size, not too heavy (like boxes, often too thick for precise mulching use. Wood chips are from when the tree service puts benches through the professional grade chipper.
But do wait at least a year or more with the gout weed fully covered , and not mixed in with the plants you want. These roots can live a long time underground, or under this kind of cover.
Good luck. And if you have time, being out with your well fitting knee pads on, picking those pesky roots out by hand, isn't the worst way to spend some down time.