1. **State the problem:** Simplify the expression \(\frac{x^2 - 9}{2x^2 - 7x - 15}\). 2. **Recall the formulas and rules:** - Difference of squares: \(a^2 - b^2 = (a - b)(a + b)\). - Factor quadratic expressions by finding two numbers that multiply to \(ac\) and add to \(b\) in \(ax^2 + bx + c\). - Simplify fractions by canceling common factors. 3. **Factor the numerator:** \[ x^2 - 9 = (x - 3)(x + 3) \] 4. **Factor the denominator:** We want to factor \(2x^2 - 7x - 15\). - Multiply \(a \times c = 2 \times (-15) = -30\). - Find two numbers that multiply to \(-30\) and add to \(-7\): these are \(-10\) and \(3\). - Rewrite the middle term: \[ 2x^2 - 10x + 3x - 15 \] - Group terms: \[ (2x^2 - 10x) + (3x - 15) \] - Factor each group: \[ 2x(x - 5) + 3(x - 5) \] - Factor out common binomial: \[ (2x + 3)(x - 5) \] 5. **Rewrite the expression with factors:** \[ \frac{(x - 3)(x + 3)}{(2x + 3)(x - 5)} \] 6. **Check for common factors:** None of the factors cancel. 7. **Final simplified expression:** \[ \frac{(x - 3)(x + 3)}{(2x + 3)(x - 5)} \]