http://en.wikipedia.org/wiki/Quantum_harmonic_oscillator
Once you noticed the relationship:
the problems with the perturbations proportional to either x^2 or x^4 are also easy to solve.
By applying the operator x to the state n, you will have the states n+1 and n-1. Therefore, if the operator is x^2, what you will have are the states n+2, n, and n-2.
I will leave it to you what you will get with the operator x^4, and compute the first and the second order energy shift for each case of the perturbations.