1. **State the problem:** Solve the expression $3 - 2(x-1)^2 + \frac{1}{2}(x+1)$ and simplify it. 2. **Recall the formulas and rules:** - Square of a binomial: $(a-b)^2 = a^2 - 2ab + b^2$ - Distributive property: $a(b+c) = ab + ac$ - Combine like terms to simplify expressions. 3. **Expand the squared term:** $$ (x-1)^2 = x^2 - 2x + 1 $$ 4. **Substitute back and distribute:** $$ 3 - 2(x^2 - 2x + 1) + \frac{1}{2}(x+1) $$ $$ = 3 - 2x^2 + 4x - 2 + \frac{1}{2}x + \frac{1}{2} $$ 5. **Combine like terms:** Constants: $3 - 2 + \frac{1}{2} = 1 + \frac{1}{2} = \frac{3}{2}$ Linear terms: $4x + \frac{1}{2}x = \frac{8}{2}x + \frac{1}{2}x = \frac{9}{2}x$ Quadratic term: $-2x^2$ 6. **Final simplified expression:** $$ -2x^2 + \frac{9}{2}x + \frac{3}{2} $$ This is the simplified form of the given expression.