Show that $\gcd(a + b, a^2 + b^2) = 1\mbox{ or} 2$

Show that $\gcd(a + b, a^2 + b^2) = 1\mbox{ or} 2$

Show that $\gcd(a + b, a^2 + b^2) = 1\mbox{ or} 2$?
I know the fact that $\gcd(a,b)=1$ implies $\gcd(a,b^2)=1$ and
$\gcd(a^2,b)=1$, but how do I apply this to that?