The question:

Show that $\displaystyle |z_1 + z_2|^2 + |z_1 - z_2|^2 = 2|z_1|^2 + 2|z_2|^2$, for all complex numbers $\displaystyle z_1$ and $\displaystyle z_2$.

I tried getting the LHS to match the RHS, but I've done something wrong. My attempt involved factorising using the difference of two squares, but it doesn't seem to help. Any assistance would be great.