Combination Chemotherapy With or Without Erlotinib Hydrochloride in Treating Patients With Metastatic or Recurrent Squamous Cell Carcinoma of the Head and Neck

Inclusion Criteria:

* Histologically confirmed metastatic or recurrent squamous cell carcinoma of the head and neck (SCCHN) of the oral cavity, oropharynx, hypopharynx or larynx; metastatic or recurrent lesions of the nasopharynx and sinus are excluded
* Radiologically measurable disease, defined as at least one lesion that can be accurately measured in at least one dimension (longest diameter to be recorded) as \>= 20 mm with conventional techniques or as \>= 10 mm with spiral computed tomography (CT) scan; measurable lymph nodes are required to be \>= 15 mm in size (short axis diameter)
* Eastern Cooperative Oncology Group (ECOG) performance status (PS) =\< 2
* Absolute neutrophil count (ANC) \>= 1.5 x 10\^9/L
* Platelet count \>= 100 x 10\^9/L
* Total bilirubin =\< upper limit of normal (ULN) (excluding Gilbert's disease)
* Alanine aminotransferase (ALT) (serum glutamate pyruvate transaminase \[SGPT\]) =\< 1.5 x ULN
* Alkaline phosphatase =\< 2.5 x ULN
* Serum creatinine =\< 1.5 x ULN
* Patients with reproductive potential (e.g., females menopausal for less than 1 year and not surgically sterilized) must practice effective contraceptive measures for the duration of study drug therapy and for at least 30 days after completion of study drug therapy; female patients of childbearing potential must provide a negative pregnancy test (serum or urine) =\< 14 days prior to treatment initiation
* Written informed consent to participate in the study according to the investigational review board (IRB) or independent ethics committee (IEC)

Exclusion Criteria:

* Histology other than squamous cell carcinoma
* Primary sites other than oral cavity, oropharynx, hypopharynx, and larynx
* Prior palliative chemotherapy for metastatic or recurrent disease
* Prior biological therapy for metastatic or recurrent disease within 3 weeks prior to randomization
* Patients with known, untreated brain metastases; patients with treated (irradiated or resected) brain metastases are eligible if treatment was completed more than 28 days prior to study entry and if clinical neurologic function is stable
* Pre-existing peripheral neuropathy \>= grade 2
* History of poorly controlled gastrointestinal disorders that could affect the absorption of the study drug (e.g., Crohn's disease, ulcerative colitis); patients requiring feeding tubes are permitted
* Other active malignancies requiring chemotherapy treatment within 2 years prior to randomization, except for adequately treated basal cell or squamous cell skin cancer or in situ cervical or breast cancer or superficial, resected melanoma
* Serious underlying medical condition which would impair the ability of the patient to receive protocol treatment, in the opinion of the treating physician
* History of allergic reactions to compounds of similar chemical composition to the study drugs (docetaxel, cisplatin, carboplatin, erlotinib or their excipients), or other drugs formulated with polysorbate 80
* Any concurrent anti-cancer therapy, excluding hormonal therapy for prostate or breast cancer
* Dementia or significantly altered mental status that would prohibit the understanding and giving of informed consent
* Women who are pregnant or breast-feeding and women or men not practicing effective birth control