1. Problem: Simplify $(-2x - x) \cdot (-2)$. Formula: Use distributive property $a(b+c) = ab + ac$. Work: $(-2x - x) = -3x$. Then $-3x \cdot (-2) = 6x$. 2. Problem: Simplify $\frac{9x^6}{x^3}$. Formula: $\frac{a^m}{a^n} = a^{m-n}$. Work: $\frac{9x^6}{x^3} = 9x^{6-3} = 9x^3$. 3. Problem: Simplify $\frac{8x^3}{4x}$. Formula: $\frac{a}{b} = \frac{8}{4} \cdot \frac{x^3}{x^1}$. Work: $\frac{8}{4} = 2$ and $\frac{x^3}{x} = x^{3-1} = x^2$. Answer: $2x^2$. 4. Problem: Expand $(x + 8)^2$. Formula: $(a+b)^2 = a^2 + 2ab + b^2$. Work: $x^2 + 2 \cdot x \cdot 8 + 8^2 = x^2 + 16x + 64$. 5. Problem: Expand $(x - 6)^2$. Formula: $(a-b)^2 = a^2 - 2ab + b^2$. Work: $x^2 - 2 \cdot x \cdot 6 + 6^2 = x^2 - 12x + 36$. 6. Problem: Simplify $(-3x + 3) \cdot (3 - 3x)$. Formula: Use distributive property. Work: $(-3x)(3) + (-3x)(-3x) + 3(3) + 3(-3x) = -9x + 9x^2 + 9 - 9x = 9x^2 - 18x + 9$. 7. Problem: Simplify $-10 \cdot (x - 5)$. Formula: Distribute $-10$. Work: $-10x + 50$. 8. Problem: Simplify $5 - 4 \cdot (3x - 3)$. Formula: Distribute $-4$. Work: $5 - 12x + 12 = 17 - 12x$. 9. Problem: Expand $(2x - 2y)^2$. Formula: $(a-b)^2 = a^2 - 2ab + b^2$. Work: $(2x)^2 - 2 \cdot 2x \cdot 2y + (2y)^2 = 4x^2 - 8xy + 4y^2$. 10. Problem: Expand $(x^2 - 3)^2$. Formula: $(a-b)^2 = a^2 - 2ab + b^2$. Work: $(x^2)^2 - 2 \cdot x^2 \cdot 3 + 3^2 = x^4 - 6x^2 + 9$. 11. Problem: Simplify $(2x^2 - 5)(2x^2 + 5)$. Formula: $(a-b)(a+b) = a^2 - b^2$. Work: $(2x^2)^2 - 5^2 = 4x^4 - 25$. Final answers: 1. $6x$ 2. $9x^3$ 3. $2x^2$ 4. $x^2 + 16x + 64$ 5. $x^2 - 12x + 36$ 6. $9x^2 - 18x + 9$ 7. $-10x + 50$ 8. $17 - 12x$ 9. $4x^2 - 8xy + 4y^2$ 10. $x^4 - 6x^2 + 9$ 11. $4x^4 - 25$