This partially randomized phase I/II trial studies the side effects and best dose of veliparib when given together with combination chemotherapy and to see how well they work in treating patients with stage IV head and neck cancer. Drugs used in chemotherapy, such as docetaxel, cisplatin, and fluorouracil, work in different ways to stop the growth of tumor cells, either by killing the cells, by stopping them from dividing, or by stopping them from spreading. Veliparib may stop the growth of tumor cells by blocking some of the enzymes needed for cell growth. It is not yet known whether combination chemotherapy is more effective when given with or without veliparib in treating head and neck cancer.