1. **State the problem:** Solve the equation $ (x - 9)(x + 2) = (x + 4)(x - 7) $ for $x$. 2. **Expand both sides:** Use the distributive property (FOIL) to expand each product. Left side: $$ (x - 9)(x + 2) = x \cdot x + x \cdot 2 - 9 \cdot x - 9 \cdot 2 = x^2 + 2x - 9x - 18 = x^2 - 7x - 18 $$ Right side: $$ (x + 4)(x - 7) = x \cdot x - 7x + 4x - 28 = x^2 - 7x + 4x - 28 = x^2 - 3x - 28 $$ 3. **Set the expanded expressions equal:** $$ x^2 - 7x - 18 = x^2 - 3x - 28 $$ 4. **Subtract $x^2$ from both sides to simplify:** $$ \cancel{x^2} - 7x - 18 = \cancel{x^2} - 3x - 28 $$ $$ -7x - 18 = -3x - 28 $$ 5. **Bring all terms involving $x$ to one side and constants to the other:** Add $7x$ to both sides: $$ -7x + 7x - 18 = -3x + 7x - 28 $$ $$ -18 = 4x - 28 $$ Add $28$ to both sides: $$ -18 + 28 = 4x - 28 + 28 $$ $$ 10 = 4x $$ 6. **Solve for $x$ by dividing both sides by 4:** $$ x = \frac{10}{4} $$ $$ x = \frac{\cancel{10}}{\cancel{4}} = \frac{5}{2} $$ **Final answer:** $$ x = \frac{5}{2} $$