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 \(b+c\).

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