1. State the problem. Problem: Multiply out and simplify the following expressions a) $x - 2(x + 2) - 2(x - 2)^2$ b) $x - 2(x + 2) - 2(x - 2)(x + 2)$ c) $(2a - 3)^2 - (2a - 3)(3a + 2)$ d) $(x/3 - 3/2)^2 - (x/3 - 3/2)(x/3 + 3/2)$. 2. Formula and rules. Use the distributive law: $a(b+c)=ab+ac$. Use the square of a binomial: $ (u - v)^2 = u^2 - 2uv + v^2$. When subtracting an expression, distribute the negative sign to each term inside the parentheses. 3. Work and final answers. 3.a) Expand and simplify step by step. $$(x-2)^2 = x^2 - 4x + 4$$ $$x - 2(x + 2) - 2(x - 2)^2 = x - 2x - 4 - 2(x^2 - 4x + 4)$$ $$= x - 2x - 4 - 2x^2 + 8x - 8$$ $$= -2x^2 + 7x - 12$$ Final answer for a): $-2x^2 + 7x - 12$. 3.b) Expand and simplify step by step. $$(x - 2)(x + 2) = x^2 - 4$$ $$x - 2(x + 2) - 2(x - 2)(x + 2) = x - 2x - 4 - 2(x^2 - 4)$$ $$= x - 2x - 4 - 2x^2 + 8$$ $$= -2x^2 - x + 4$$ Final answer for b): $-2x^2 - x + 4$. 3.c) Expand squares and products and simplify. $$(2a - 3)^2 = 4a^2 - 12a + 9$$ $$(2a - 3)(3a + 2) = 6a^2 - 5a - 6$$ $$(2a - 3)^2 - (2a - 3)(3a + 2) = 4a^2 - 12a + 9 - (6a^2 - 5a - 6)$$ $$= 4a^2 - 12a + 9 - 6a^2 + 5a + 6$$ $$= -2a^2 - 7a + 15$$ Final answer for c): $-2a^2 - 7a + 15$. 3.d) Factor a common term and simplify using cancellation. $$ (x/3 - 3/2)^2 - (x/3 - 3/2)(x/3 + 3/2) = (x/3 - 3/2)\left((x/3 - 3/2) - (x/3 + 3/2)\right)$$ $$= (x/3 - 3/2)\left(\cancel{x/3} - \cancel{x/3} - 3/2 - 3/2\right)$$ $$= (x/3 - 3/2)(-3)$$ $$= -3\cdot \frac{x}{3} + \frac{9}{2}$$ $$= -\frac{3x}{3} + \frac{9}{2}$$ $$= -\frac{\cancel{3}x}{\cancel{3}} + \frac{9}{2}$$ $$= -x + \frac{9}{2}$$ Final answer for d): $-x + 9/2$.