If $a^n + 1$ is prime for some numbers $a \geq 2$ and $n\geq 1$, show that n must be a power of 2.
I understand how to do this proof. Specifically by showing that $x^n + 1$ is composite. My question is how to come up with the notion to show that $(x+1)$ divides $x^n + 1$. My prof used a hand wavy argument stating that because $x^m + 1$ equals 0 when $x = -1$ But that does not comvince me in the slightest. I am not seeing how that implies divisibility.