Prove that if $x$ is odd then $x^2 + (x + 2)^2$ is divisible by $2$ but not for $4$ This is what I tried:
$$x^2 + x^2 + 4x + 4=2x^2 + 4(x + 1),$$
so it's divisible by $2$, since this expression is a sum of a multiple of $2$ and a multiple of $4$.
Therefore, for the expression not to be a multiple of $4$, $2x^2$ can't be multiple of $4$, that's what I can't prove.
Brazilian student, sorry for my English.

HINT:
$$x^2+(x+2)^2=\{(x+2)-x\}^2+2x(x+2)$$
As $x$ is odd $\iff x+2$ is odd, so will be $x(x+2)$
