Answer: What you describe sounds like powdery mildew, a fungal disease. Your Bougainvillea may be under stress due to insufficient sunlight. Couple that with dry air (common indoors during the winter months), and poor air circulation and you can understand how your plant might develop disease problems. There's not much you can do to control the disease other than removing the affected plant parts and changing the growing environment so your plant can recover. Your bougainvillea should spend the summer months outdoors so pruning now (in early spring) will get it ready for a trip out into the garden. When you bring it back indoors next autumn, be sure to provide direct sunlight (or supplement natural light with fluorescent lighting) and give your plant a shower every few weeks. This will rinse off any dust and discourage insect pests. It will also increase humidity around the plant (and make it think it experienced a rainshower).
Best wishes with your bougainvillea!
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