1. The problem is to factor the polynomial $P(x) = x^3 + 7x^2 - 9x - 63$ given that $(x + 7)$ is a factor. 2. According to the Factor Theorem, if $(x + 7)$ is a factor, then $P(-7) = 0$. 3. Perform synthetic division of $P(x)$ by $(x + 7)$: $$ -7 \;|\; 1 \quad 7 \quad -9 \quad -63 \\ \quad \downarrow \quad -7 \quad 0 \quad 63 \\ \quad 1 \quad 0 \quad -9 \quad 0 $$ 4. The quotient polynomial is $x^2 - 9$. 5. Factor $x^2 - 9$ using the difference of squares formula: $$x^2 - 9 = (x - 3)(x + 3)$$ 6. Therefore, the full factorization of $P(x)$ is: $$P(x) = (x + 7)(x - 3)(x + 3)$$ This means the roots of $P(x)$ are $x = -7, 3, -3$.