We report the first application of the Schwinger multichannel formulation to low-energy electron collisions with a nonlinear polyatomic target. Integral and differential cross sections are obtained for e-CH4 collisions from 3 to 20 eV at the static-plus-exchange interaction level. In these studies the exchange potential is directly evaluated and not approximated by local models. An interesting feature of the small-angle differential cross section is ascribed to polarization effects and not reproduced at the static-plus-exchange level. Our differential cross sections are found to be in reasonable agreement with existing measurements at 7.5 eV and higher energies.