Docetaxel, Cisplatin and Fluorouracil in Treating Patients With Previously Untreated Stage II-IV Nasal Cavity and Paranasal Sinus Cancer

Inclusion Criteria: * Patients must have a confirmed (by a MD Anderson Cancer Center \[MDACC\] pathologist) cytologic or histological diagnosis of locally advanced squamous cell carcinoma, poorly differentiated carcinoma, or sinonasal undifferentiated carcinoma of the nasal cavity and/or paranasal sinuses. * Stage II-IV disease; tumor (T) 2-4, node (N) any, metastasis (M) 0. Measurable disease is required with the following criteria: Measurable lesions can be accurately measured, with at least one diameter \>= 1.0 cm by spiral computed tomography (CT) scan or magnetic resonance imaging (MRI). Lesions can be bidimensionally measurable or unidimensionally measurable. Every effort should be made to measure lesions in two dimensions. Measurable disease is present if the patient has one or more measurable lesions. Non-measurable lesions/disease are all other lesions, including small lesions (those with measurements \< 2.0 cm; or \< 1.0 cm with spiral CT). * Eastern Cooperative Oncology Group (ECOG) Performance Status (PS) 0-1. * Absolute peripheral granulocyte count (AGC) of \>= 1500 cells/mm\^3. * Platelet count of \>= 100,000 cells/mm\^3. * Total bilirubin =\< upper limit of normal (ULN). If the patient has a history of Gilbert's Syndrome, check direct and indirect bilirubin. If in the judgment of the attending medical oncologist it is safe to treat the patient, the patient will be considered eligible for this criteria. * Alkaline phosphatase =\< 2 x ULN. If in the judgment of the attending medical oncologist it is safe to treat the patient, the patient will be considered eligible for this criteria. * Aspartate aminotransferase (AST) and alanine aminotransferase (ALT) =\< 2 x ULN. If in the judgment of the attending medical oncologist it is safe to treat the patient, the patient will be considered eligible for this criteria. * Hemoglobin \>= 10.0g/dL. * Per MDACC creatinine clearance (CrCl) guidelines, patients must have a creatinine clearance \> 50 ml/min determined by 24 hour collection or nomogram * Patients should have uncontrolled intercurrent illness, which in the opinion of the attending medical oncologist, would render the patient unsuitable for the study (i.e., preclude safe administration of the prescribed chemotherapy treatment). * Women of child bearing potential (WOCBP) must have a negative serum or urine pregnancy test (i.e., minimum sensitivity 25 IU/L or equivalent units of human chorionic gonadotropin \[HCG\]), within 72 hours prior to the start of study treatment. Should a woman become pregnant or suspect she is pregnant while participating in the study, she should inform her treating physician(s) immediately. * Ability to understand and the willingness to sign a written Informed Consent Document (ICD). In the event that non-English speaking participants are eligible for this study, a short form (if applicable) or an ICD in their language, will be utilized and completed in accordance with the MDACC Policy For Consenting Non-English Speaking Participants. * Willingness to undergo MDACC Audiology and Ophthalmology Assessment. Exclusion Criteria: * Evidence of distant metastases (below the clavicle) by clinical or radiographic measures. * Pre-existing peripheral neuropathy Common Terminology Criteria for Adverse Events (CTCAE) grade 2 or worse. * Pre-existing bilateral sensorineural hearing loss at \> 90dB at any frequency from 250-8000Hz as assessed by a comprehensive audiometric evaluation for patients receiving cisplatin. This criteria will not apply to patients receiving carboplatin. * Prior chemotherapy (i.e., as administered strictly for cancer treatment) within the previous 3 years. Use of chemotherapy agents for non-cancer treatment purposes (i.e., arthritis treatment, etc.) are excluded from this criterion. * Prior radiotherapy to the paranasal sinus region or the upper neck (i.e., prior radiotherapy to another disease site is acceptable). * Initial surgical resection of the paranasal sinuses or nasal cavity region rendering the patient clinically and radiologically disease free. * Simultaneous primary invasive cancers or patients currently receiving any other investigational agents at time of study enrollment. Patients may have received investigational agents in the past. No washout time period is required. * Patients with a past history of malignancy that were treated less than 3 years and have not remained disease free for the past 3 years. (Patients with non metastatic skin cancers will be eligible). * Men and women of childbearing potential (WOCBP) who are unwilling or unable to use an acceptable method to avoid pregnancy for the entire study period and for up to 3 months after the study. Subjects who are men must also agree to use effective contraception. Note: WOCBP must be using an adequate method of contraception throughout the study and for up to 3 months after the study. Adequate methods of contraception will include (oral, injectable, or implantable hormonal contraceptive; tubal ligation; intra-uterine device; barrier contraceptive with spermicide; or vasectomized partner). * Women who are pregnant or breastfeeding. * Patients with a history of severe hypersensitivity reaction to docetaxel or other drugs formulated with polysorbate 80. * Patients with a known history of human immunodeficiency virus (HIV).