1. **State the problem:** Solve the fractional equation $$\frac{2x}{x-3} + \frac{4}{x+2} = \frac{10}{x^2 - x - 6}$$. 2. **Factor the denominator on the right side:** Note that $$x^2 - x - 6 = (x-3)(x+2)$$. 3. **Rewrite the equation with factored denominator:** $$\frac{2x}{x-3} + \frac{4}{x+2} = \frac{10}{(x-3)(x+2)}$$ 4. **Find the least common denominator (LCD):** The LCD is $$(x-3)(x+2)$$. 5. **Multiply both sides of the equation by the LCD to clear denominators:** $$\left(\frac{2x}{x-3} + \frac{4}{x+2}\right)(x-3)(x+2) = \frac{10}{(x-3)(x+2)}(x-3)(x+2)$$ 6. **Simplify each term:** - $$\frac{2x}{x-3} \times (x-3)(x+2) = 2x(x+2)$$ - $$\frac{4}{x+2} \times (x-3)(x+2) = 4(x-3)$$ - Right side simplifies to $$10$$ 7. **Write the simplified equation:** $$2x(x+2) + 4(x-3) = 10$$ 8. **Expand the terms:** $$2x^2 + 4x + 4x - 12 = 10$$ 9. **Combine like terms:** $$2x^2 + 8x - 12 = 10$$ 10. **Bring all terms to one side:** $$2x^2 + 8x - 12 - 10 = 0$$ $$2x^2 + 8x - 22 = 0$$ 11. **Divide the entire equation by 2 to simplify:** $$x^2 + 4x - 11 = 0$$ 12. **Use the quadratic formula to solve for $x$:** $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ where $a=1$, $b=4$, $c=-11$. 13. **Calculate the discriminant:** $$\Delta = 4^2 - 4 \times 1 \times (-11) = 16 + 44 = 60$$ 14. **Find the roots:** $$x = \frac{-4 \pm \sqrt{60}}{2} = \frac{-4 \pm 2\sqrt{15}}{2} = -2 \pm \sqrt{15}$$ 15. **Check for restrictions:** The denominators cannot be zero, so $x \neq 3$ and $x \neq -2$. 16. **Final solution:** $$x = -2 + \sqrt{15} \quad \text{or} \quad x = -2 - \sqrt{15}$$ Both values do not equal 3 or -2, so both are valid. **Answer:** $$x = -2 \pm \sqrt{15}$$