You can root a bougainvillea from a cutting if you dip the cutting in rooting hormone and place it in a moist potting mix with a clear cover over it to hold in moisture. Put this somewhere with good bright light but not direct sun. Don't take the tender tip of a shoot but instead use a section back from the tip that is not so tender and succulent. The following web site has a section that offers more info on rooting cuttings.
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