Dear Student,
Honeysuckle:Cutting::Bougainvillea:stem cutting
The easiest way of propagation is cutting, take a cutting from your bougainvillea, this is the part of the plant that isn't brand new, but isn't established and overly woody. Remove any leaves from the cutting and insert upright in a mix of one perlite and one part peat. Sink it in one or two inches into the growing medium.
Thank You!