REGIMEN I: Patients receive paclitaxel intravenously (IV) over 3 hours, carboplatin IV over 30 minutes, and bevacizumab IV over 30-90 minutes (beginning in course 2) on day 1. Patients also receive veliparib orally (PO) twice daily (BID) on days 1-21. Treatment repeats every 21 days for 6 courses. Patients then receive bevacizumab alone on day 1. Treatment with bevacizumab repeats every 21 days for 16 courses in the absence of disease progression or unacceptable toxicity.