1. **State the problem:** Simplify the expression $\frac{x^2 - 9}{5x^2 - 11x - 12}$.\n\n2. **Recall the formula and rules:** To simplify a rational expression, factor numerator and denominator completely and then cancel common factors.\n\n3. **Factor the numerator:** $x^2 - 9$ is a difference of squares, so $x^2 - 9 = (x - 3)(x + 3)$.\n\n4. **Factor the denominator:** Factor $5x^2 - 11x - 12$. Find two numbers that multiply to $5 \times (-12) = -60$ and add to $-11$. These are $-15$ and $4$.\n\nRewrite the middle term: $5x^2 - 15x + 4x - 12$.\n\nGroup terms: $(5x^2 - 15x) + (4x - 12)$.\n\nFactor each group: $5x(x - 3) + 4(x - 3)$.\n\nFactor out common binomial: $(x - 3)(5x + 4)$.\n\n5. **Rewrite the expression:** $\frac{(x - 3)(x + 3)}{(x - 3)(5x + 4)}$.\n\n6. **Cancel common factors:** $(x - 3)$ cancels out, leaving $\frac{x + 3}{5x + 4}$.\n\n**Final answer:** $$\frac{x + 3}{5x + 4}$$