I'm not sure what this bush is called, but it appears that one side of it isn't leafing out. Can anybody offer some insight on this?
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I believe this plant is a Daphne x burkwoodii "Carol Mackie". It's evergreen in many zones, and has small fragrant flowers in the early spring. It's a very valuable garden plant but can be severely affected by winter cold and especially winds. It also likes even moisture and a partly shady position in your garden.
The good news is that you can cut it back, even severely, in my experience (by as much as 2/3rds), to create new growth in the dead-looking areas. The best time to cut it back is in late spring after it flowers, but you could also do it any time before fall.
It looks like a good-sized, established specimen. Daphne do not like to be moved, so leave it where it is.
I've really enjoyed having this shrub in my garden - I love the variegated foliage and the sweet-smelling spring flowers.