1. **State the problem:** Simplify the expression and solve the equation given by $$k - 8 = 8$$ 2. **Simplify the polynomial expression:** Given: $$(x + 1)(x - 2)(x + 2)^2 (x - 1) - (x + 2)^3$$ Step-by-step simplification: $$= (x + 1)(x - 2)(x + 2)^2 - (x + 2)^3$$ $$= (x + 2)^2 (x + 1)(x - 2) - (x + 2)^3$$ $$= (x + 2)^2 [(x + 1)(x - 2) - (x + 2)]$$ Expand inside the bracket: $$(x + 1)(x - 2) = x^2 - 2x + x - 2 = x^2 - x - 2$$ So, $$= (x + 2)^2 [x^2 - x - 2 - (x + 2)]$$ $$= (x + 2)^2 [x^2 - x - 2 - x - 2]$$ $$= (x + 2)^2 (x^2 - 2x - 4)$$ 3. **Note:** The user’s simplification had a small error in the last step; the correct expression inside the bracket is $x^2 - 2x - 4$, not $x^2 - 4$. 4. **Solve the equation:** Given: $$k - 8 = 8$$ Add 8 to both sides: $$k - 8 + 8 = 8 + 8$$ $$\cancel{k - 8} + 8 = 16$$ So, $$k = 16$$ **Final answer:** $$k = 16$$