1. Problem statement: Factorize $1 - 2ab - (a^2 + b^2)$. 2. Expand the parentheses to combine like terms: $$1 - 2ab - (a^2 + b^2) = 1 - 2ab - a^2 - b^2$$ 3. Recognize that $a^2 + 2ab + b^2$ is a perfect square and rewrite the expression as a difference of squares: $$1 - 2ab - a^2 - b^2 = 1 - (a^2 + 2ab + b^2) = 1 - (a + b)^2$$ 4. Apply the difference of squares formula $x^2 - y^2 = (x - y)(x + y)$ with $x=1$ and $y=a+b$, then simplify: $$1 - (a + b)^2 = (1 - (a + b))(1 + (a + b)) = (1 - a - b)(1 + a + b)$$ 5. Final answer: $(1 - a - b)(1 + a + b)$.