Can you plant a cut rose stem and grow a rose bush?

Yes, you can. Most commercial roses are grown as grafted stems on briar rose root stock, but it is quite possible to grow roses from cuttings. Your local garden shop can supply rooting compound and some advice on the best way to do it.

Interestingly, your rose may have some intellectual property controls on it, and it may be illegal to grow your own copies, rather like its illegal to copy a DVD. Don't try to sell any roses you succeed in growing.