1. **State the problem:** Simplify the expression $$\frac{1 - x}{1 + x} + (1 - x)^2$$ and verify if the given simplification $$1 - x^2 + 1 + x^2$$ is correct. 2. **Recall the formulas and rules:** - To add expressions, find a common denominator if necessary. - Expand squares using the formula $$(a - b)^2 = a^2 - 2ab + b^2$$. 3. **Start with the original expression:** $$\frac{1 - x}{1 + x} + (1 - x)^2$$ 4. **Expand the square:** $$(1 - x)^2 = 1^2 - 2 \cdot 1 \cdot x + x^2 = 1 - 2x + x^2$$ 5. **Rewrite the expression:** $$\frac{1 - x}{1 + x} + 1 - 2x + x^2$$ 6. **Find a common denominator to combine terms:** Rewrite $1 - 2x + x^2$ as $$\frac{(1 - 2x + x^2)(1 + x)}{1 + x}$$ to have the same denominator. 7. **Multiply numerator:** $$(1 - 2x + x^2)(1 + x) = 1(1 + x) - 2x(1 + x) + x^2(1 + x) = (1 + x) - 2x - 2x^2 + x^2 + x^3 = 1 + x - 2x - 2x^2 + x^2 + x^3 = 1 - x - x^2 + x^3$$ 8. **Rewrite the entire expression with common denominator:** $$\frac{1 - x}{1 + x} + \frac{1 - x - x^2 + x^3}{1 + x} = \frac{(1 - x) + (1 - x - x^2 + x^3)}{1 + x} = \frac{1 - x + 1 - x - x^2 + x^3}{1 + x}$$ 9. **Combine like terms in numerator:** $$1 - x + 1 - x - x^2 + x^3 = 2 - 2x - x^2 + x^3$$ 10. **Final simplified expression:** $$\frac{2 - 2x - x^2 + x^3}{1 + x}$$ 11. **Check the given simplification:** The user wrote $$1 - x^2 + 1 + x^2 = 2$$ which simplifies to 2, but this does not match the correct simplification. **Answer:** The given simplification is incorrect. The correct simplified form is $$\frac{2 - 2x - x^2 + x^3}{1 + x}$$.