Question: top-center: fraction expression with numerator $-4x^3 (2x + 2) - 2x^2 (4x + 4)$ and denominator $2x^2$ appearing near the upper center of the image.
1. **State the problem:** Simplify the expression $$\frac{-4x^3 (2x + 2) - 2x^2 (4x + 4)}{2x^2}$$.
2. **Expand the numerator:**
$$-4x^3 (2x + 2) = -4x^3 \cdot 2x - 4x^3 \cdot 2 = -8x^4 - 8x^3$$
$$-2x^2 (4x + 4) = -2x^2 \cdot 4x - 2x^2 \cdot 4 = -8x^3 - 8x^2$$
3. **Combine the expanded terms:**
$$-8x^4 - 8x^3 - 8x^3 - 8x^2 = -8x^4 - 16x^3 - 8x^2$$
4. **Rewrite the expression:**
$$\frac{-8x^4 - 16x^3 - 8x^2}{2x^2}$$
5. **Factor the numerator:**
$$-8x^2 (x^2 + 2x + 1)$$
6. **Rewrite the expression with factored numerator:**
$$\frac{-8x^2 (x^2 + 2x + 1)}{2x^2}$$
7. **Cancel common factors:**
$$\frac{\cancel{-8x^2} (x^2 + 2x + 1)}{\cancel{2x^2}} = -4 (x^2 + 2x + 1)$$
8. **Recognize perfect square:**
$$x^2 + 2x + 1 = (x + 1)^2$$
9. **Final simplified expression:**
$$-4 (x + 1)^2$$