This is a Phase Ib/II, single center, open label, dose-finding study to evaluate the use of Belinostat when given with standard of care chemotherapy in patients with untreated, non-small cell lung cancer (NSCLC). In the Phase Ib portion, dose limiting toxicity evaluation will be used to determine the maximum tolerated dose (MTD) of Belinostat when given with fixed doses of bevacizumab, carboplatin, and paclitaxel(a BelCap-B regimen). Three dose levels of Belinostat are proposed (600mg/kg, 800mg/kg, 1000mg/kg). Determination of MTD will be the basis for establishing set dosing for the phase II component of the study.