A major problem in physics is the identification of the mechanism of high-temperature superconductivity. Many theories purport to explain the high-Tc phenomenon. In most pictures, the high-energy spin dynamics are of key importance. There has been some remarkable experimental progress in this area in last two years. Magnetic resonant inelastic x-ray spectroscopy (RIXS) has shown that highly-damped magnetic excitations exist in number of superconducting cuprate systems with energies of 0.25eV. We propose to extend our neutron scattering measurements on optimally doped La2-xSrxCuO4 into the energy range measured by RIXS. The neutron scattering measurements will be complementary to RIXS which can only access a limited momentum range and does not have a well established scattering cross section. Our results will feed into theories of high temperature superconductivity.