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 induction chemotherapy (ICT) for women of childbearing potential * Women of childbearing potential and male participants are counseled on birth control and must agree to use a medically effective means of birth control throughout their participation in the treatment phase of the study (until at least 60 days following the last study treatment) * Patient must sign study specific informed consent prior to study entry Exclusion Criteria: * Prior invasive malignancy (except non-melanomatous skin cancer) unless disease free for a minimum of 3 years * Patients with simultaneous primaries or bilateral tumors are excluded * Patients who have had initial surgical treatment other than the diagnostic biopsy of the primary site or nodal sampling of the neck disease are excluded * Patients with unknown primary tumor sites are excluded * Patients who present with a cervical lymph node metastasis of unknown primary origin * Prior systemic chemotherapy for the study cancer; note that prior chemotherapy for a different cancer is allowable * Prior radiotherapy that would result in overlap of radiation therapy fields * Primary site of tumor of oral cavity, nasopharynx, nasal cavity, paranasal sinuses, or salivary glands * Recurrent head and neck cancer * Current uncontrolled cardiac disease; i.e., uncontrolled hypertension, unstable angina, recent myocardial infarction (within prior 6 months), uncontrolled congestive heart failure, and cardiomyopathy with decreased ejection fraction * Congestive heart failure with left ventricular ejection fraction \< 20% * Transmural myocardial infarction within the last 6 months * Acute bacterial or fungal infection requiring intravenous antibiotics at registration * Chronic obstructive pulmonary disease exacerbation or other respiratory illness requiring hospitalization or precluding study therapy at the time of registration * Active lupus erythematosus or scleroderma with ongoing physical manifestations * Any uncontrolled condition, which in the opinion of the investigator, would interfere in the safe and timely completion of study procedures * Pregnant or lactating women or women of childbearing potential and men who are sexually active and not willing/able to use medically acceptable forms of contraception * Prior allergic reaction to the study drug(s) involved in this protocol * Patient is enrolled in another investigational trial