What Factor Theorem Do You Know?

Algebra Level 3

$\large P(x) =x^3+2x^2 + 2x + c \qquad\qquad Q(x) = x^2 + bx + b$

Consider the polynomials $P(x)$ and $Q(x)$ above, where $b$ and $c$ are constant real numbers and $c\ne0$ .

It is known that $Q(x)$ is a factor of $P(x)$ . Find the value of $b+c$ .