1. **State the problem:** Solve the equation $ (x - 5)(x + 4) = (x + 7)(x - 6) $. 2. **Use the distributive property (FOIL) to expand both sides:** $$ (x - 5)(x + 4) = x^2 + 4x - 5x - 20 = x^2 - x - 20 $$ $$ (x + 7)(x - 6) = x^2 - 6x + 7x - 42 = x^2 + x - 42 $$ 3. **Set the expanded expressions equal:** $$ x^2 - x - 20 = x^2 + x - 42 $$ 4. **Subtract $x^2$ from both sides to simplify:** $$ \cancel{x^2} - x - 20 = \cancel{x^2} + x - 42 $$ $$ -x - 20 = x - 42 $$ 5. **Add $x$ to both sides:** $$ -x + x - 20 = x + x - 42 $$ $$ -20 = 2x - 42 $$ 6. **Add 42 to both sides:** $$ -20 + 42 = 2x - 42 + 42 $$ $$ 22 = 2x $$ 7. **Divide both sides by 2:** $$ \frac{22}{\cancel{2}} = \frac{2x}{\cancel{2}} $$ $$ 11 = x $$ **Final answer:** $$ x = 11 $$