I had a long stemmed rose in a bouquet of flowers. The head had died so I went to discard the stem, only to notice new leaves were shooting out of it in 3 different places. I left the stem in the vase of water and came here to find out what to do next.
You can try rooting it if you want by snipping the stem just below a bud at the base, stripping off the thorns and leaves, leaving one leaf at the top if you like, but you can take them all off, and inserting it into a sharp sand or sand/compost mix in a deep pot. The stem should be around 9 inches long, and you need to bury it so that only a quarter of the stem is showing. If the thorns don't snap off easily, then just bin it, its too old. Otherwise, water and wait, but this procedure is usually done in summer, outdoors, and you'd expect the cuttings to have rooted by about November - yours may not be successful because its been cut so long and you are obliged to carry out the treatment inside because the stem isn't hardened off.
Bamboo's answer is good. I agree with everything stated there, just wanted to add the method I used when I did this years ago.
I recut the cut ends (flat, and clean), but then topped the stems off (they were really long, so removed about half), leaving them at 9-10 inches. I cut about 1/4" above a large healthy leaf. Then I removed the leaves halfway up the stem, dipped the stripped (lower) half in fungicide/rooting hormone powder, and inserted them into moist, gritty sterile mix. This was placed in a high-humidity case under bright filtered light, and I used a bottom heat of 75 degrees Fahrenheit.
One turned black and I removed it, but the rest (11) rooted in about 3 - 4 weeks (slower than I expected), and began to grow. So I moved them to 1 gal. nursery pots of standard potting mix, and began fertilizing and spraying black-spot preventative. This was in early spring (late March, I think).
They were ready for sale 2 years later. I determined (through some sleuthing) that I had the 'First Red' production rose. It's a hybrid tea, bred by Paul Pekmez in France, in 1988, and was patented (US patent No: PP 7,749 on 17 Dec 1991), so I just gave them away, so as not to run into trouble.
It was basically just a project carried out for fun, and made great gifts. Now, it needs protection in my area (hardy only in zones 7b up), so that's something you should expect if you get some started, even though you'll likely have a different production rose than I did. I ran into no pest or disease problems out of the ordinary for garden roses (surprising to me), and what came up (mostly Japanese beetles) was easily taken care of.
I bought this rose at rite aid on clearance two years ago. We put it in water, and the stem never turned brown. After about 9 months in a glass of water, I put it in soil. It grew a bunch of new branches, and about two weeks ago I saw a bud. I took this photo this morning. So, for people who say it can't re-root or grow new flowers, they are wrong! This is proof.
Pioneer women traveling West on wagons transplanted rose cuttings by inserting them into slits made in common potatoes. If they survived that and grew into rose bushes at the end of the trip, many rose cuttings are likely to survive and grow. Just make fresh cuttings by cutting roses from bouquets much shorter and including a few growth joints.