1. **State the problem:** Write the expression $ (x - 1)^2 - x(x + 1)(x - 3) $ as a standard polynomial. 2. **Expand each part:** - Expand $ (x - 1)^2 $ using the formula $ (a - b)^2 = a^2 - 2ab + b^2 $: $$ (x - 1)^2 = x^2 - 2x + 1 $$ - Expand $ x(x + 1)(x - 3) $ step-by-step: First, expand $ (x + 1)(x - 3) $: $$ (x + 1)(x - 3) = x^2 - 3x + x - 3 = x^2 - 2x - 3 $$ Then multiply by $ x $: $$ x(x^2 - 2x - 3) = x^3 - 2x^2 - 3x $$ 3. **Rewrite the original expression:** $$ (x - 1)^2 - x(x + 1)(x - 3) = (x^2 - 2x + 1) - (x^3 - 2x^2 - 3x) $$ 4. **Distribute the minus sign:** $$ x^2 - 2x + 1 - x^3 + 2x^2 + 3x $$ 5. **Combine like terms:** - Combine $ x^3 $ terms: $ -x^3 $ - Combine $ x^2 $ terms: $ x^2 + 2x^2 = 3x^2 $ - Combine $ x $ terms: $ -2x + 3x = x $ - Constant term: $ 1 $ 6. **Final polynomial:** $$ -x^3 + 3x^2 + x + 1 $$ This is the expression written as a standard polynomial.