Nonhomogeneous Equations & Resonance · Superchallenge problem

A damped spring obeys $x''+2x'+5x=4\cos\omega t$. In the steady state, the forcing delivers average power $\overline{P}(\omega)$, the time average over one period of $4\cos(\omega t)\cdot x'(t)$. Find the largest value of $\overline{P}(\omega)$ over all $\omega>0$.