I need to prove the parallelogram law for complex numbers z and w: $\displaystyle |z+w|^2+|z-w|^22|z|^2+2|w|^2$. My approach follows:

Proof: $\displaystyle |z+w|^2+|z-w|^2=|z|^2+2|wz|+|w|^2+|z|^2-2|wz|+|w|^2=2|z|^2+2|w|^2$

Can I actually multiply |z+w| in this format? Thanks.