1. **State the problem:** Simplify the expression $$(x - 2)^2 + 3(x + 1)^3 - (x + 9)$$. 2. **Recall formulas:** - Square of a binomial: $$(a - b)^2 = a^2 - 2ab + b^2$$ - Cube of a binomial: $$(a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3$$ 3. **Expand each term:** - Expand $$(x - 2)^2$$: $$x^2 - 2 \cdot x \cdot 2 + 2^2 = x^2 - 4x + 4$$ - Expand $$(x + 1)^3$$: $$x^3 + 3x^2 \cdot 1 + 3x \cdot 1^2 + 1^3 = x^3 + 3x^2 + 3x + 1$$ 4. **Multiply the cube expansion by 3:** $$3(x^3 + 3x^2 + 3x + 1) = 3x^3 + 9x^2 + 9x + 3$$ 5. **Rewrite the expression with expansions:** $$x^2 - 4x + 4 + 3x^3 + 9x^2 + 9x + 3 - (x + 9)$$ 6. **Distribute the minus sign in the last term:** $$x^2 - 4x + 4 + 3x^3 + 9x^2 + 9x + 3 - x - 9$$ 7. **Combine like terms:** - Combine $x^3$ terms: $$3x^3$$ - Combine $x^2$ terms: $$x^2 + 9x^2 = 10x^2$$ - Combine $x$ terms: $$-4x + 9x - x = 4x$$ - Combine constants: $$4 + 3 - 9 = -2$$ 8. **Final simplified expression:** $$3x^3 + 10x^2 + 4x - 2$$