I have to admit I haven't been successful growing roses from cuttings, though I have tried a few times. My problem has always been that they do grow new side shoots, but fail to grow roots and the shoots get moldy and turn black, OR even when they do grow roots and fairly healthy side shoots, I haven't been able to keep them alive long enough.
Next time I try, I'm going to follow the techniques described here in this thread
Subject: Help with growing roses from cuttings?
It's hard to tell what variety this rose is, especially since its petals have been dyed with colored water. So it may not be the recommended easy-er to grow from cuttings variety. But you could try and see. However, hybrid teas are the typical cut flowers with the classic shape, and they are notorious for not being able to grow strong roots of their own that can support themselves (they are grafted to stronger rootstock as a standard practice)