Patients who consent to this study will receive treatment for four 28 day cycles unless they are unable to continue study treatment due to toxicity or disease progression. Paclitaxel and carboplatin will be given intravenously on day 1 of each cycle and galunisertib in pill form on days 4-17. No medicine will be given on days 17-28 of each cycle. Patients may have the option of continuing treatment with the study drug if they complete the 4 cycles the study requires. Subjects will receive tests and procedures that are part of regular cancer care as well as those required for the purposes of the study. Blood samples will be taken to monitor the level of study drug in the blood. Patients will have the option of allowing blood and tumor samples to be used for future research.