Combination Chemotherapy With or Without Veliparib in Treating Patients With Stage IV Head and Neck Cancer

Inclusion Criteria: * PHASE I: * Patients who are treatment naïve, high risk, stage IVa/IVb (all other sites) and histologically proven squamous cell carcinoma of the head and neck (SCCHN) with no definitive evidence of metastatic disease, excluding patients with oropharynx human papillomavirus (HPV)-positive tumors; in summary, those patients eligible are newly diagnosed and treatment naive: * Stage IVa-b squamous cell carcinoma other than oropharyngeal cancer (OPC), or * Oropharyngeal cancer (OPC) HPV-negative, stage IVa-b * PHASE II: * Patients who are treatment naïve, high risk, stage IVa/IVb (all other sites) histologically proven SCCHN with no definitive evidence of metastatic disease; in summary, those patients eligible are: * Stage IVa-b SCCHN other than OPC, or * OPC, HPV-negative, IVa-b, or * OPC, HPV positive, with greater than 10 pack-year smoking history and N2b-N3 disease * PHASE I AND II: * Patients must have at least one measurable site of disease according to Response Evaluation Criteria in Solid Tumors (RECIST) criteria; i.e., patients must have measurable disease, defined as at least one lesion that can be accurately measured in at least one dimension (longest diameter to be recorded for non-nodal lesions and short axis for nodal lesions) as \>= 20 mm with conventional techniques or as \>= 10 mm with spiral computed tomography (CT) scan magnetic resonance imaging (MRI), or calipers by clinical exam * Eastern Cooperative Oncology Group (ECOG) performance status 0-1 * Patients must be able to swallow the drug * Ability to understand and the willingness to sign a written informed consent document * Leukocytes \>= 3,000/mm\^3 * Absolute neutrophil count \>= 1,500/mm\^3 * Platelets \>= 100,000/mm\^3 * Total bilirubin =\< 1.5 institutional upper limit of normal (ULN) * Serum glutamic oxaloacetic transaminase (SGOT) (aspartate aminotransferase \[AST\]) and serum glutamate pyruvate transaminase (SGPT) (alanine aminotransferase \[ALT\]) =\< 2.5 x institutional ULN as calculated by Cockcroft-Gault * Creatinine clearance \>= 60 mL/min/1.73 m\^2 for patients with creatinine levels above ULN as calculated by Cockcroft-Gault * Patients who are receiving any other investigational agents are not eligible * Patients with active seizure or a history of seizure are not eligible * Patients with a history of allergic reactions attributed to compounds of similar chemical or biologic composition to ABT-888 or other agents used in study, including Cremophor, carboplatin, paclitaxel, cisplatin, 5-fluorouracil, hydroxyurea, or any compounds of similar chemical or biologic composition are not eligible * Patients with impairment of gastrointestinal function or gastrointestinal disease that may significantly alter the absorption of ABT-888 (e.g., ulcerative disease, uncontrolled nausea, vomiting, diarrhea, malabsorption syndrome or small bowel resection) are not eligible to participate in this study * Patients with uncontrolled intercurrent illness including, but not limited to, ongoing or active infection, symptomatic congestive heart failure, unstable angina pectoris, cardiac arrhythmia, or psychiatric illness/social situations that would limit compliance with study requirements are not eligible to participate in the study * Pregnant women are not eligible to participate in this study; NOTE: women of child bearing potential must have a negative serum or urine pregnancy test within 7 days prior to treatment * Women of child-bearing potential and men must agree to use adequate contraception (hormonal or barrier method of birth control; abstinence) prior to study entry and for the duration of study participation; should a woman become pregnant or suspect she is pregnant while she or her partner is participating in this study, she should inform her treating physician immediately; * Breastfeeding should be discontinued if the mother is treated with ABT-888 * Human immunodeficiency virus (HIV)-positive patients on combination antiretroviral therapy are not eligible * Patients receiving chronic, systemic treatment with corticosteroids or another immunosuppressive agent are not eligible to participate in this study; topical or inhaled corticosteroids are allowed * Patients with other malignancies within the past 2 years, except for adequately treated carcinoma of the cervix or basal or squamous cell carcinomas of the skin or surgically treated early stage solid tumors are ineligible to participate in this study