Help with a divides 4 proof For any integer $k$, $k^2 + 2k + 3$ is even if and only if $4| k^2 − 2k − 7$.
I know that I will have to use the rule that if $2|n^2$
for some integer $n$, then $4|n^2$. 
Does anyone know if I need to use a contrapositive or converse proof? 

Both are equivalent to $2|k+1$:
$$
2|(k+1)^2+2\iff 2|(k+1)^2\iff 2|k+1,
$$
$$
4|(k-1)^2-8\iff4|(k-1)^2\iff2|k-1\iff2|k+1.
$$
