1. **State the problem:** Evaluate the expression $$(1 - w + w^2)(1 + w - w^2).$$ 2. **Expand the expression:** Use the distributive property (FOIL) to multiply each term in the first parenthesis by each term in the second parenthesis. $$(1)(1) + (1)(w) + (1)(-w^2) + (-w)(1) + (-w)(w) + (-w)(-w^2) + (w^2)(1) + (w^2)(w) + (w^2)(-w^2)$$ 3. **Simplify each term:** $$1 + w - w^2 - w - w^2 + w^3 + w^2 + w^3 - w^4$$ 4. **Combine like terms:** Notice $w$ and $-w$ cancel out: $$1 + (w - w) + (-w^2 - w^2 + w^2) + (w^3 + w^3) - w^4 = 1 + (-w^2) + 2w^3 - w^4$$ Simplify the $w^2$ terms: $$1 - w^2 + 2w^3 - w^4$$ 5. **Final answer:** $$oxed{1 - w^2 + 2w^3 - w^4}$$