You will get suggestions for all sorts of home remedies. For mealy bug, you
will need a systemic insecticide, either Orthenex or Di-Syston granules.
Neither of them is disastrously dangerous if you use them correctly. Check the
label carefully to make sure it is OK for Bougainevillea or its close relatives
(four-o-clock family). Spray with Orthenex in the garage or an unused room, &
leave the plant there for a couple of hours.
Central NY, Zone 5a, Sunset Zone 40
"A tree never hits an automobile except in self defense." - Woody Allen
Repeat application of Orthene, read and follow all label directions.