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