1. **State the problem:** Solve and simplify the expression $$(1-x)^2 + 2x(1-x)$$. 2. **Recall formulas and rules:** - The square of a binomial: $$(a-b)^2 = a^2 - 2ab + b^2$$. - Distributive property: $$a(b+c) = ab + ac$$. 3. **Expand the first term:** $$ (1-x)^2 = 1^2 - 2 \cdot 1 \cdot x + x^2 = 1 - 2x + x^2 $$ 4. **Expand the second term:** $$ 2x(1-x) = 2x \cdot 1 - 2x \cdot x = 2x - 2x^2 $$ 5. **Combine the expanded terms:** $$ (1 - 2x + x^2) + (2x - 2x^2) $$ 6. **Simplify by combining like terms:** $$ 1 - 2x + x^2 + 2x - 2x^2 = 1 + \cancel{-2x + 2x} + (x^2 - 2x^2) $$ 7. **Cancel terms and simplify:** $$ 1 + 0 + (x^2 - 2x^2) = 1 - x^2 $$ **Final answer:** $$ 1 - x^2 $$