Cisplatin, Intensity-Modulated Radiation Therapy, and Pembrolizumab in Treating Patients With Stage III-IV Head and Neck Squamous Cell Carcinoma

Inclusion Criteria: * STEP 1 (REGISTRATION) * Pathologically (histologically or cytologically) proven diagnosis of head and neck squamous cell carcinoma (HNSCC) involving the oral cavity (excluding lips), oropharynx (p16 negative), hypopharynx or larynx * Patients must have undergone gross total surgical resection of high-risk oral cavity, oropharynx (p16 negative), larynx, or hypopharynx squamous cell carcinoma (SCC) within 63 days prior to registration; note: patients may have a biopsy under general anesthesia in an operating room followed by definitive ablative cancer surgery representing gross total resection; the gross total resection has to be done within 63 days prior to registration; if, however, patients have ablative resection but demonstrate rapid gross recurrence or are determined to have gross persisting disease requiring re-resection to achieve gross total resection, then the patient is not eligible * Patients must have at least one of the following high risk pathologic features: * Extracapsular nodal extension * Invasive cancer at the primary tumor resection margin (tumor on ink); Note: Patients who have a positive margin and undergo re-resection with final negative margin are eligible only if they can be enrolled within 63 days of initial gross total resection AND extracapsular nodal extension was also present; patients who have a positive margin and undergo re-resection with final negative margin and do not have extracapsular nodal extension, are NOT el…