1. **State the problem:** Simplify and analyze the quadratic expression $-4x^2 - 32x - 28$. 2. **Formula and rules:** To simplify, factor out the greatest common factor (GCF) from all terms. 3. **Find the GCF:** The coefficients are -4, -32, and -28. The GCF is -4. 4. **Factor out the GCF:** $$-4x^2 - 32x - 28 = -4(x^2 + 8x + 7)$$ 5. **Factor the quadratic inside the parentheses:** Find two numbers that multiply to 7 and add to 8, which are 7 and 1. 6. **Write the factorization:** $$-4(x + 7)(x + 1)$$ 7. **Explanation:** Factoring helps us find the roots of the quadratic by setting each factor equal to zero. 8. **Find roots:** $$x + 7 = 0 \Rightarrow x = -7$$ $$x + 1 = 0 \Rightarrow x = -1$$ 9. **Summary:** The expression factors to $-4(x + 7)(x + 1)$ with roots at $x = -7$ and $x = -1$. This is a downward-opening parabola because the coefficient of $x^2$ is negative (-4).