TRYHARD: Radiation Therapy Plus Cisplatin With or Without Lapatinib in Treating Patients With Head and Neck Cancer.

Inclusion criteria: * Patients must have histologically or cytologically confirmed diagnosis (from primary lesion and/or lymph nodes) of Squamous Cell Cancer of the oropharynx, hypopharynx or larynx (For patients with oropharynx primary, the tumor must be negative for p16 by immunohistochemistry). * Patients with selected Stage III or IV disease (T2 N2-3 M0, T3-4 any N M0, T1 N2b, N2c or N3 p16 negative oropharynx cancer or T1-2 any N+ hypopharynx cancer) including no distant metastases. * History/Physical examination by a Radiation Oncologist and Medical oncologist prior to entering the study. * Examination by an ears, nose, throat (ENT) or Head \& Neck Surgeon including laryngopharyngoscopy prior to entering the study. * Patients must have a chest CT scan, or positron emission tomography (PET)/CT scan to rule out metastatic disease * Patients must have a contrast enhanced CT scan or MRI or PET/CT scan of the tumor site and neck nodes prior to entering the study. * Patients must have an EKG and echocardiogram (ECHO) or multigated acquisition (MUGA) scan prior to entering the study. * Patients must have Zubrod Performance Status of 0-1. * Patients must be ≥ 18 years of age. * Patients must have normal organ and marrow function as defined below: * Absolute neutrophil count (ANC) ≥ 1,500 cells/mm3 * Platelets ≥ 100,000 cells/mm3 * Hemoglobin ≥ 8.0 g/dl * Serum creatinine \< 1.5 mg/dl or creatinine clearance (CC) ≥ 50 ml/min * Total bilirubin \< 2 x the institutional upper limit of normal * Aspartate aminotransferase (AST) or alanine aminotransferase (ALT) ≤ 3 x the institutional upper limit of normal * Patient must have magnesium, calcium, glucose, potassium and sodium levels within normal limits * Women of childbearing potential must have a negative pregnancy test prior to registration. * Patients of reproductive potential must practice effective contraception while on study and for at least 60 calendar days following treatment. * All patients must sign an informed consent prior to enrollment. * Patients must comply with the treatment plan and follow-up schedule. Exclusion criteria: * Patients with simultaneous primaries or bilateral tumors. * Patients who have had gross total excision of the primary tumor. * Patients with initial surgical treatment, radical or modified neck dissection. * Patients who received prior systemic chemotherapy for the study cancer. * Patients who received prior radiotherapy to the region of the study cancer that would result in overlap of radiation therapy fields. * Patients with primary tumor of oral cavity, nasopharynx, sinuses or salivary glands. * Prior allergic reaction to the study drugs. * Patients who have had prior therapy that specifically and directly targets the epidermal growth factor receptor (EGFR)/human epidermal growth factor receptor 2 (HER2) pathway. * Patients who have current active hepatic or biliary disease (with exception of patients with Gilbert's syndrome, asymptomatic gallstones, or stable chronic liver disease per investigator assessment); * Pregnant women or sexually active patients not willing or able to use medically acceptable forms of contraceptive method while on treatment. * Patients with severe, active co-morbidity, defined as follows: * Uncontrolled cardiac disease, such as uncontrolled hypertension, unstable angina, and/or congestive heart failure requiring hospitalization within the last 6 months * Transmural myocardial infarction within the last 6 months * Left ventricular ejection fraction \< 45% * Acute bacterial or fungal infection requiring intravenous antibiotics at the time of registration * Chronic Obstructive Pulmonary Disease exacerbation or other respiratory illness requiring hospitalization or precluding study therapy within 30 calendar days prior to registration * Hepatic insufficiency resulting in clinical jaundice and/or Coagulation defects * Acquired Immune Deficiency Syndrome (AIDS) based upon current Centers for Disease Control and Prevention (CDC) definition