Show that for the harmonic oscillator with Hamiltonian H₀=p²/2m+1/2mω²x² subject to the potential V=αx³+βx⁴, the correction to the first-order energy in perturbation theory is E₁=3/2β(ℏ²/m²ω²)(n²+n+1), with n=0,1,2,…