The purpose of this study is to test the effectiveness (how well the drug works), safety, and tolerability of the investigational drug combination of palbociclib (Ibrance) plus carboplatin in patients with metastatic head and neck squamous cell cancer.

Inclusion Criteria:

• Histologically documented progressive squamous cell head and neck cancer with or without metastases, not amenable to curative treatment; or the patient has documented refusal of curative treatment.
• ECOG performance status of 0-2. Eastern Cooperative Oncology Group Performance Status: an attempt to quantify cancer patients' general well-being and activities of daily life. The score ranges from 0 to 5 where 0 is asymptomatic and 5 is death.
• Presence of measurable disease by CT scan per RECIST v1.1.
• Age ≥18 years.
• Life expectancy of ≥12 weeks.
• Women of childbearing potential must have a negative serum or urine pregnancy test at time of screening and confirmed within 3 days prior to treatment. Women not of child-bearing potential will be defined as all women older than age 50 and anovulatory for 12 months.
• Signed and dated informed consent document indicating that the patient (or legally acceptable representative) has been informed of all pertinent aspects of the trial prior to enrollment.
• Willingness and ability to comply with scheduled visits, treatment plans, laboratory tests, and other study procedures.
• Adequate organ and marrow function

Exclusion Criteria:

• Previous treatment with cytotoxic chemotherapy therapy in the recurrent/metastatic setting. Previous treatment with non-cytotoxic agents in the recurrent/metastatic setting is permitted. Gastrointestinal abnormalities causing impaired absorption precluding administration of oral medications.
• Evidence of untreated or progressive brain metastases, spinal cord compression, or carcinomatous meningitis.
• A serious uncontrolled medical disorder or active infection that would impair their ability to receive study treatment.
• Dementia or significantly altered mental status that would prohibit the understanding or rendering of informed consent and compliance with the requirements of this protocol.
• Patients (male and female) having procreative potential who are not willing or not able to use adequate contraception. Women who are pregnant or breast-feeding.
• Patients residing in prison.
• Prior experimental therapy within 30 days of enrollment.
• Availability of curative treatment option for the patient's cancer, whether surgery, chemotherapy, radiation, or combination thereof, unless the patient has documented refusal of curative treatment.
• Current use or anticipated inability to avoid use of drugs that are known strong CYP3A4/5 inhibitors (atazanavir, boceprevir, conivaptan, clarithromycin, grapefruit or grapefruit juice, indinavir, itraconazole, ketoconazole, nelfinavir, nefazodone, posaconazole, ritonavir, saquinavir, telaprevir, telithromycin, voriconazole ).
• Current use or anticipated inability to avoid use of drugs that are known strong CYP3A4/5 inducers (carbamazepine, dexamethasone, fosphenytoin, phenytoin, phenobarbital, rifabutin, rifampin, rifapentine, St. John's wort).
• Patients with a history of severe allergic reaction to cisplatin or carboplatin