If $x^2+bx+b$ is a factor of $x^3+2x^2+2x+c$, $c \neq 0$, then $b-c=?$

Question

I am not getting where to start! Please help me out

Answer 1

$$(x^2+bx+b)(x+\alpha)=x^3 + 2x^2 + 2x +c$$ $$x^3 + (b+\alpha)x^2 +b(\alpha+1)x + b\alpha = x^3 + 2x^2 +2x+c$$

So we have:

$$b+\alpha=2$$

$$b(\alpha+1)=2$$

$$b\alpha =c$$

Solving for $\alpha$ we get $\alpha(1-\alpha)=0$.

We cannot have $\alpha=0$ because then $c$ would be zero, so $\alpha=1$, $b=1$, $c=1$.

$$b-c=0.$$