Question
This may be out of your expertise.  I got a small piece of my neighbor's rosemary bush, with a small piece of root attached.  Their bush is very large and spreading, and is actually flowering at this time, small lavender flowers.  I would like to successfully root it myself, either in my house or outside, much like my neighbor's.  I do not know what variety of rosemary it is; I don't think they even do.  Can you offer some suggestions for me to  get this piece of rosemary to successfully root, and become a healthy plant?  We live in zone 7b, mid-North Carolina.  Thank you.

Answer
The main thing I noticed in your description is, "a small piece of root attached." If in fact it has a root, you are half way there.
Put some sterile growing medium (potting soil) in a very small pot. Very carefully put the little root in a hole you have made in the soil, and gently press the soil around it. The root will break off easily, so it is important that you are careful at this stage.
Place the pot in a bright window. Let the soil dry between watering. By spring it should be rooted enough to plant out side.
I suggest the indoor method, rather then planting it directly outside, because it has a much better chance of survival if the roots are established before you put it in the outdoor soil.
bakerplanter