Paclitaxel and Carboplatin Before Radiation Therapy With Paclitaxel in Treating HPV-Positive Patients With Stage III-IV Oropharynx, Hypopharynx, or Larynx Cancer

Inclusion Criteria: * Pathologically (histologically or cytologically) proven (from primary lesion and/or lymph nodes) diagnosis of HPV-positive squamous cell carcinoma of the oropharynx, hypopharynx, or larynx; HPV-positivity will be defined as tumors that are p16-positive by immunohistochemistry * Clinical stage III or IV disease; note: patients with M1 tumors are not eligible * Appropriate stage for protocol entry, including no distant metastases, based upon the following minimum diagnostic workup: * History/physical examination within 4 weeks prior to registration, including assessment of weight loss in past 6 months * Chest x-ray (or chest computed tomography \[CT\] scan or positron emission tomography \[PET\]/CT scan) within 6 weeks prior to registration * CT scan or magnetic resonance imaging (MRI) of the head and neck (of the primary tumor and neck nodes) and PET/CT scan * Zubrod performance status 0-1 * Absolute neutrophil count (ANC) \> 1,800 cells/mm\^3 * Platelets \> 100,000 cells/mm\^3 * Hemoglobin (Hgb) \> 8.0 g/dl (note: the use of transfusion or other intervention to achieve Hgb \> 8.0 g/dl is acceptable) * Aspartate aminotransferase (AST) or alanine aminotransferase (ALT) \< 2x the upper limit of normal * Serum creatinine =\< 1.5 mg/dl or institutional upper limit of normal * Creatinine clearance (CC) \>= 50 ml/min determined by 24-hour collection or estimated by Cockcroft-Gault formula * Negative serum pregnancy test within 7 days prior to start of ind…