Completing the square of $(m^2 + n^2)$

Question

I'm attempting to complete the square of $m^2 + n^2$, How would I do this? I am not understanding, as most resources refer to a polynomial with $x$ as it's variable and every term is in terms of $x$.

Edit: I'm trying to do this in a proof. I'm trying to prove if $m-n$ is even, then $m^2 - n^2$ is even. My lecturer in class said that it may be easier to complete the square of the following. All I wrote down was $2*(n^2 + mn)$. Where can I go from there?

Answer 1

Found the answer independently, Completing the square was the phrase my professor used, but I just needed (m^2 + n^2) in a different form.

  (m + n)^2 - 2mn = (m^2 + n^2)

Just expand (m + n)^2 and it becomes apparent.

Answer 2

When you draw a square of $n \times n$ and a square of $m \times m$ where they share one corner point, you will see that you are missing two pieces of $m \times n$ to complete an $(m+n) \times (m+n)$ square.