1. The problem is to analyze and simplify the quadratic expression $$-4x^2 + 24x - 4$$. 2. First, factor out the greatest common factor from all terms, which is $$-4$$. 3. Factoring out $$-4$$ gives: $$-4(x^2 - 6x + 1)$$ 4. Next, complete the square inside the parentheses to express the quadratic in vertex form. 5. Complete the square for $$x^2 - 6x + 1$$: Add and subtract $$\left(\frac{6}{2}\right)^2 = 9$$ inside the parentheses: $$x^2 - 6x + 9 - 9 + 1 = (x - 3)^2 - 8$$ 6. Substitute back: $$-4((x - 3)^2 - 8)$$ 7. Distribute the $$-4$$: $$-4(x - 3)^2 + 32$$ 8. This is the vertex form of the quadratic, where the vertex is at $$(3, 32)$$. 9. This shows the quadratic opens downward (since the coefficient of the squared term is negative) and has a maximum point at the vertex. Final answer: $$-4(x - 3)^2 + 32$$