A Study of Alisertib and Paclitaxel in Patients With Small Cell Lung Cancer (SCLC)

The goal of this clinical trial is to determine how well people tolerate treatment with alisertib at different doses when it is used together with paclitaxel to treat people with SCLC. The main question it aims to answer is: * What percentage of side effects, both mild and serious, do participants experience when being treated with alisertib and paclitaxel based on the National Cancer Institute Common Terminology Criteria for Adverse Events version 5.0 (NCI CTCAE v.5.0)? The study will consist of different groups, called cohorts, in which alisertib will be studied at increasing doses. Participants in the first group, Cohort 1, will take 30 mg of alisertib by mouth 2 times a day. The dose will increase by 10 mg 2 times a day for each new cohort of participants joining the study. Side effects will be checked during the study, and the decision to increase the dose of alisertib will be based on the specific side effects experienced during the first 21 days of treatment for each cohort. Participants will: * take alisertib by mouth on their own 2 times a day from Day 1 through Day 7 of each 21-day cycle and will be given paclitaxel intravenously at a dose of 60 mg/m2 on Day 1 and Day 8 of each 21-day cycle. * be given a preventive treatment to increase their body defenses, called granulocyte-colony stimulating factor (G-CSF). Study staff will give G-CSF after the last dose of alisertib or paclitaxel of each 21-day cycle in which alisertib was given. * provide blood samples to evaluate the different levels of alisertib in the blood at different times. Blood samples will be collected on Days 1 and 7 of the first and second 21-day cycles of treatment. * be treated with alisertib and paclitaxel until death, worsening of the disease, unacceptable toxicity, unwillingness to continue participating in the trial, or other criteria that requires participants to stop treatment and participation in the study. * be contacted every 8 weeks after stopping alisertib and paclitaxel treatment as part of a specific study phase called the long-term follow-up period. This will last until death, unwillingness to continue participating in the trial, or until the end of the study.