1. **State the problem:** Solve the equation $$\frac{x}{x^2 - 4x - 12} - \frac{1}{x+2} = \frac{5}{x+2}$$ for $x$. 2. **Factor the quadratic denominator:** $$x^2 - 4x - 12 = (x - 6)(x + 2)$$ 3. **Rewrite the equation with factored denominator:** $$\frac{x}{(x - 6)(x + 2)} - \frac{1}{x + 2} = \frac{5}{x + 2}$$ 4. **Find the common denominator:** The common denominator is $(x - 6)(x + 2)$. 5. **Rewrite each term with the common denominator:** $$\frac{x}{(x - 6)(x + 2)} - \frac{1 \cdot (x - 6)}{(x + 2)(x - 6)} = \frac{5 \cdot (x - 6)}{(x + 2)(x - 6)}$$ 6. **Combine the left side:** $$\frac{x - (x - 6)}{(x - 6)(x + 2)} = \frac{5(x - 6)}{(x - 6)(x + 2)}$$ 7. **Simplify the numerator on the left:** $$x - (x - 6) = x - x + 6 = 6$$ So the equation becomes: $$\frac{6}{(x - 6)(x + 2)} = \frac{5(x - 6)}{(x - 6)(x + 2)}$$ 8. **Since denominators are equal and not zero, set numerators equal:** $$6 = 5(x - 6)$$ 9. **Solve for $x$:** $$6 = 5x - 30$$ $$6 + 30 = 5x$$ $$36 = 5x$$ $$x = \frac{36}{5}$$ 10. **Check for restrictions:** Denominators cannot be zero, so $x \neq 6$ and $x \neq -2$. Our solution $x = \frac{36}{5} = 7.2$ is valid. **Final answer:** $$x = \frac{36}{5}$$