Sorafenib in Combination With Carboplatin and Paclitaxel in Treating Participants With Metastatic or Recurrent Head and Neck Squamous Cell Cancer

Inclusion Criteria: * Ability to understand and the willingness to sign a written informed consent. A signed informed consent must be obtained prior to any study specific procedures. * Patients must have cytologically or histologically proven recurrent or metastatic squamous cell cancer of the head and neck (SCCHN) from the primary tumor or lymph nodes of the oral cavity, larynx, oropharynx, or hypopharynx. * No prior systemic chemotherapy for patients who present with metastatic disease. For patients with recurrent head and neck squamous cell carcinoma, prior chemotherapy is allowed if it was given as part of their definitive therapy. If patients have received prior combined modality therapy, they must be off therapy for at least 6 months. * Patients must have at least 1 evaluable lesion. Lesions must be evaluated by computed Tomography (CT) scan or magnetic resonance imaging (MRI). * Eastern Cooperative Oncology Group (ECOG) performance status (PS) of 0-1. * Controlled blood pressure (defined as systolic blood pressure \[BP\] =\< 140 mmHg and diastolic =\< 85 mmHg). * Hemoglobin \>= 9.0 g/dL within 7 days prior to start of first dose. * Absolute neutrophil count (ANC) \>= 1,500/mm\^3 within 7 days prior to start of first dose. * Platelet count \>= 100,000/mm\^3 within 7 days prior to start of first dose. * Total bilirubin =\< 1.5 times the upper limit of normal (ULN) within 7 days prior to start of first dose. * Alanine aminotransferase (ALT) and aspartate aminotransferase (AST) =\< 2.5 x ULN (=\< 5 x ULN for patients \[pts\] with \[w/\] liver involvement) within 7 days prior to start of first dose. * International normalized ratio (INR) =\< 1.5 and partial thromboplastin time (PTT) within (w/in) normal limits within 7 days prior to start of first dose. * Serum creatinine =\< 1.5 ULN or creatinine clearance (CrCl) \>= 45 mL/min for patients (pts) w/ creatinine levels above institutional normal within 7 days prior to start of first dose. * Amylase \& lipase \< 1.5 x the ULN within 7 days prior to start of first dose. * Urinalysis (UA) must show less than 1+ protein in urine, or the pt will require a repeat UA. If repeat UA shows 1+ protein or more, a 24 hour urine collection will be required \& must show total protein =\< 1000 mg/24 hour to be eligible. * Women of childbearing potential (not surgically sterilized or at least 2 years postmenopausal) must have a negative serum or urine pregnancy test performed within 7 days prior to the start of treatment. * Women of childbearing potential and men must agree to use adequate contraception (abstinence; hormonal or barrier method of birth control) prior to study entry, for the duration of study participation and 3 months after end of treatment. Should a woman become pregnant while participating or the partner of a patient participating in this study becomes pregnant, they should inform their treating physician immediately. Exclusion Criteria: * Congestive heart failure (CHF) \> class II New York Heart Association (NYHA); active coronary artery disease (myocardial infarction \[MI\] more than 6 months prior to study entry is allowed); or serious cardiac ventricular arrhythmias requiring anti-arrhythmic therapy (beta blockers or digoxin are permitted). * Uncontrolled hypertension defined as systolic blood pressure \> 140 mmHg or diastolic pressure \> 85 mmHg despite optimal medical management. * Known history of human immunodeficiency virus (HIV) infection or chronic hepatitis B or C. * Active clinically serious infections (i.e. patients currently taking antibiotics) (grade 2 National Cancer Institute \[NCI\]-Common Terminology Criteria for Adverse Events (CTCAE) version 3.0). * Evidence or history of central nervous system (CNS) disease, including primary brain tumors, seizures disorders, or any brain metastasis. * Thrombotic or embolic events such as cerebrovascular accident, deep vein thrombosis or pulmonary embolism. History of transient ischemic attack is allowed. * Evidence or history of bleeding diathesis or coagulopathy. * History of/or current evidence of hemoptysis (bright red blood of ½ teaspoon or more). * Peripheral neuropathy \>= grade 2 (NCI-CTC version 3.0). * Anticancer chemotherapy or immunotherapy: anticancer therapy is defined as any agent or combination of agents with clinically proven anticancer activity administered by any route with the purpose of affecting the cancer, either directly or indirectly, including palliative and therapeutic endpoints. * Radiotherapy to the target lesions within 3 weeks of start of first dose. Toxicities from radiotherapy must have resolved prior to start of first dose. * No major surgery, open biopsy or significant traumatic injury within 4 weeks of start of first dose. * Serious, non-healing wound, ulcer, or bone fracture. * Granulocyte growth factors (G-CSF), within 3 weeks of study entry. * Patients taking chronic erythropoietin are permitted provided no dose adjustment is made within 2 months prior to start of first dose. * Pregnant or breastfeeding patients. * Substance abuse, medical, psychological or social conditions that may interfere with the patient's participation in the study or evaluation of the study results. * Known or suspected allergy to any recombinant human antibodies, or compounds of similar chemical or biologic composition to sorafenib or any of the drugs in this study. * Any condition that is unstable or could jeopardize the safety or compliance of the patient in the study. * Previous or concurrent cancer that is distinct in primary site or histology from the cancer being evaluated in the study EXCEPT cervical cancer in situ, treated basal cell carcinoma, superficial bladder tumors (Ta, Tis and T1) or any cancer curatively treated \> 3 years prior to study entry. * Known or suspected allergy to sorafenib (BAY 43-9006) or any agent given in association with this trial. * Any malabsorption conditions. * Therapeutic anticoagulation with warfarin, heparins, or heparinoids. * Patients taking phenytoin, carbamazepine, and phenobarbital. * Patients taking rifampin and/or St. John's Wort. * Patients who are candidates for curative surgery or radiotherapy. * The patient has progressed within 6 months after completion of curative intent (definitive) treatment for localized/locoregionally advanced disease.