The \(La_{1-x} Sr_x MnO_3\) system with \(0.2 \lesssim x \lesssim 0.4\) has traditionally been modelled with a ``double exchange'' Hamiltonian, in which it is assumed that the only relevant physics is the tendency of carrier hopping to line up neighboring spins. We present a solution of the double exchange model, show it is incompatible with many aspects of the resistivity data, and propose that a strong electron-phonon interaction arising from a Jahn-Teller splitting of the outer Mn d-level plays a crucial role.