1. **State the problem:** Expand and simplify the expressions: a) $ (x - 1)(x + 4) $ b) $ (x - 3)(x - 4) $ c) $ (x + 8)(x + 9) $ 2. **Formula used:** To expand two binomials, use the distributive property (FOIL method): $$ (a + b)(c + d) = ac + ad + bc + bd $$ 3. **Expand and simplify each:** a) $ (x - 1)(x + 4) = x \cdot x + x \cdot 4 - 1 \cdot x - 1 \cdot 4 $ $$ = x^2 + 4x - x - 4 $$ Combine like terms: $$ = x^2 + \cancel{4x} - \cancel{x} - 4 = x^2 + 3x - 4 $$ b) $ (x - 3)(x - 4) = x \cdot x - x \cdot 4 - 3 \cdot x + 3 \cdot 4 $ $$ = x^2 - 4x - 3x + 12 $$ Combine like terms: $$ = x^2 - \cancel{4x} - \cancel{3x} + 12 = x^2 - 7x + 12 $$ c) $ (x + 8)(x + 9) = x \cdot x + x \cdot 9 + 8 \cdot x + 8 \cdot 9 $ $$ = x^2 + 9x + 8x + 72 $$ Combine like terms: $$ = x^2 + \cancel{9x} + \cancel{8x} + 72 = x^2 + 17x + 72 $$ **Final answers:** a) $ x^2 + 3x - 4 $ b) $ x^2 - 7x + 12 $ c) $ x^2 + 17x + 72 $