If $\alpha,a,b$ are integers and $b\neq-1$, then prove that, if $\alpha$ satisfies the equation $x^2+ax+b+1=0$, $a^2+b^2$ must be composite. 
Let $\alpha,a,b$ be integers such that $b\neq-1$. Assume that $\alpha$ satisfies the equation $x^2+ax+b+1=0$. Prove that the integer $a^2+b^2$ must be composite.

$\alpha=\frac{-a\pm\sqrt{a^2-4(b+1)}}{2}$is an integer. But manipulating this expression is leading me to nowhere. Please help.

We have 
\begin{eqnarray*}
\alpha^2 + a \alpha +b+1=0.
\end{eqnarray*}
Multiply by $4$ and add $a^2$
\begin{eqnarray*}
(2 \alpha + a)^2 =a^2 -4(b+1).
\end{eqnarray*}
Now add & subtract $b^2$ & rearrange to
\begin{eqnarray*}
a^2+b^2=(2 \alpha + a+b+2)(2 \alpha + a-b-2).
\end{eqnarray*}
