1. **State the problem:** Simplify or analyze the quadratic expression $-4x^2 + 16x + 84$. 2. **Identify the formula and rules:** This is a quadratic expression of the form $ax^2 + bx + c$ where $a = -4$, $b = 16$, and $c = 84$. 3. **Factor the quadratic:** First, factor out the greatest common factor (GCF) from all terms. $$-4x^2 + 16x + 84 = -4(x^2 - 4x - 21)$$ 4. **Factor the quadratic inside the parentheses:** Find two numbers that multiply to $-21$ and add to $-4$. These numbers are $-7$ and $3$ because $-7 \times 3 = -21$ and $-7 + 3 = -4$. 5. **Write the factored form:** $$-4(x - 7)(x + 3)$$ 6. **Explain:** The expression is factored completely as $-4(x - 7)(x + 3)$. This shows the roots of the quadratic are $x = 7$ and $x = -3$. **Final answer:** $$-4(x - 7)(x + 3)$$