1. **State the problem:** Simplify the expression $\frac{x^2 - 9}{5x^2 - 11x - 12}$. 2. **Factor numerator and denominator:** - Numerator: $x^2 - 9$ is a difference of squares, so $x^2 - 9 = (x - 3)(x + 3)$. - Denominator: Factor $5x^2 - 11x - 12$. 3. **Factor denominator:** We look for two numbers that multiply to $5 \times (-12) = -60$ and add to $-11$. These numbers are $-15$ and $4$. Rewrite middle term: $$5x^2 - 15x + 4x - 12$$ Group terms: $$(5x^2 - 15x) + (4x - 12)$$ Factor each group: $$5x(x - 3) + 4(x - 3)$$ Factor out common binomial: $$(5x + 4)(x - 3)$$ 4. **Rewrite the expression:** $$\frac{(x - 3)(x + 3)}{(5x + 4)(x - 3)}$$ 5. **Cancel common factors:** Since $x - 3$ appears in numerator and denominator (and $x \neq 3$ to avoid division by zero), cancel it: $$\frac{x + 3}{5x + 4}$$ 6. **Final answer:** $$\boxed{\frac{x + 3}{5x + 4}}$$ This is the simplified form of the original expression.