Physicians are conducting a clinical trial for patients with pediatric sarcoma. Sarcoma is a type of bone cancer that can be treated with chemotherapy which can affect the bone marrow, where blood cells are produced. Neutrophils are a type of white blood cells that fight infection and are produced in the bone marrow. If the neutrophil count becomes too low due to chemotherapy, a potentially serious condition called neutropenia occurs. Neutropenia is serious because it can affect the body's ability to protect against many types of infections. Pegfilgrastim is an investigational drug being evaluated for its potential ability to increase the number of neutrophils. The purpose of this study is to determine the safety and effectiveness of pegfilgrastim in preventing neutropenia following chemotherapy in pediatric patients up through the age of 21 with sarcoma.