Previous research with a similar therapy regimen resulted in high rates of cancer shrinkage, high rates of avoiding radiation and its side effects, high cure rate and good quality of life. Radiation can be very toxic. The purpose on this study is to try to avoid radiation. If the participants are not on this study they would be receiving radiation as it is standard treatment of their cancer. In the last study with a similar regimen, about a third of cancers had a pathologic complete response with the first part of the study. This means that the chemotherapy had killed the cancer. The investigators are trying to improve the regimen further with a goal of increasing this rate of complete response to the first part of therapy. The investigators also hope that by improving results in the first part, that more people will be cured and that long term quality of life (especially speech and swallowing) will be improved, both compared to standard therapies and to the last study. Doctors do not know how this therapy will effect the participants. There is no guarantee that this study will benefit the participants.