1. **State the problem:** Simplify the expression $ (x - 3)(x + 1) + 2x(x^2 - 2x) $. 2. **Use the distributive property (FOIL) to expand each product:** $$ (x - 3)(x + 1) = x \cdot x + x \cdot 1 - 3 \cdot x - 3 \cdot 1 = x^2 + x - 3x - 3 $$ Simplify the middle terms: $$ x^2 + \cancel{x} - \cancel{3x} - 3 = x^2 - 2x - 3 $$ 3. Expand the second product: $$ 2x(x^2 - 2x) = 2x \cdot x^2 - 2x \cdot 2x = 2x^3 - 4x^2 $$ 4. Combine the two results: $$ (x^2 - 2x - 3) + (2x^3 - 4x^2) = 2x^3 + x^2 - 4x^2 - 2x - 3 $$ Simplify like terms: $$ 2x^3 + \cancel{x^2} - 4x^2 - 2x - 3 = 2x^3 - 3x^2 - 2x - 3 $$ 5. **Final simplified expression:** $$ \boxed{2x^3 - 3x^2 - 2x - 3} $$