1. Stated problem: Simplify and evaluate the expression $ (a - 1)^2 - (a + 1)(a - 1) $. 2. Use the formulas: - Square of a binomial: $ (x - y)^2 = x^2 - 2xy + y^2 $ - Product of sum and difference: $ (x + y)(x - y) = x^2 - y^2 $ 3. Expand each part: $$ (a - 1)^2 = a^2 - 2a + 1 $$ $$ (a + 1)(a - 1) = a^2 - 1 $$ 4. Substitute back into the expression: $$ (a - 1)^2 - (a + 1)(a - 1) = (a^2 - 2a + 1) - (a^2 - 1) $$ 5. Simplify by removing parentheses: $$ a^2 - 2a + 1 - a^2 + 1 $$ 6. Cancel $a^2$ terms: $$ \cancel{a^2} - 2a + 1 - \cancel{a^2} + 1 = -2a + 2 $$ 7. Final simplified expression: $$ \boxed{-2a + 2} $$ This means the original expression simplifies to $-2a + 2$.