A Study of Prexasertib (LY2606368) With Chemotherapy and Radiation in Participants With Head and Neck Cancer

Inclusion Criteria: * Must be able to comply with the treatment plan and follow-up schedule * Must have diagnosis of head and neck squamous cell carcinoma of the oropharynx (tonsil, base of tongue, soft palate, or oropharyngeal walls), hypopharynx, or larynx * Have adequate organ function * If participant is of reproductive potential, must agree to use medically approved contraceptive precautions during the study and for six months following the last dose of study drug * If the participant is a female of childbearing potential, must have had a negative serum or urine pregnancy test within 14 days of the first dose of study drug and must not be breast feeding Exclusion Criteria: * Must not have taken an unapproved drug as treatment for any indication within the last 28 days prior to starting study treatment * Must not have received any of the following prior therapies or treatments: systemic therapy for the study cancer, radiation therapy to the head and neck region, or curative-intent surgery in the head and neck region * Have evidence of a distant metastatic disease * Must not have an active symptomatic fungal, bacterial or viral infection, including human immunodeficiency virus (HIV) or Hepatitis A, B, or C * Must not have a serious heart condition, such as congestive heart failure, unstable angina pectoris, or heart attack within the last three months * Must not have a family history of long corrected QT interval (QTc) syndrome * Must not have known allergic reaction against any of the components of the study treatment