1. **State the problem:** Simplify the expression $z(y + 7) - 8(y + 7)$. 2. **Identify the common factor:** Both terms contain the factor $(y + 7)$. 3. **Use the distributive property:** Factor out $(y + 7)$ from both terms: $$z(y + 7) - 8(y + 7) = (y + 7)(z - 8)$$ 4. **Final simplified expression:** $$\boxed{(y + 7)(z - 8)}$$ This is the simplified form of the original expression by factoring out the common binomial factor.