1. **State the problem:** Solve for $x$ in the equation: $$\frac{2x - 10}{x + 4} + 10 = \frac{-2}{x - 3} - 11$$ 2. **Rewrite the equation:** $$\frac{2x - 10}{x + 4} + 10 = \frac{-2}{x - 3} - 11$$ 3. **Bring all terms to one side to combine:** $$\frac{2x - 10}{x + 4} + 10 + 11 = \frac{-2}{x - 3}$$ which simplifies to $$\frac{2x - 10}{x + 4} + 21 = \frac{-2}{x - 3}$$ 4. **Multiply both sides by the common denominator $(x+4)(x-3)$ to clear fractions:** $$\left(\frac{2x - 10}{x + 4} + 21\right)(x + 4)(x - 3) = \frac{-2}{x - 3}(x + 4)(x - 3)$$ Simplify each term: $$\left(2x - 10 + 21(x + 4)\right)(x - 3) = -2(x + 4)$$ 5. **Expand inside the parentheses:** $$2x - 10 + 21x + 84 = 23x + 74$$ So the left side is: $$(23x + 74)(x - 3)$$ 6. **Expand the left side:** $$23x^2 - 69x + 74x - 222 = 23x^2 + 5x - 222$$ 7. **Right side is:** $$-2x - 8$$ 8. **Set equation:** $$23x^2 + 5x - 222 = -2x - 8$$ 9. **Bring all terms to one side:** $$23x^2 + 5x - 222 + 2x + 8 = 0$$ Simplify: $$23x^2 + 7x - 214 = 0$$ 10. **Use quadratic formula:** $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ where $a=23$, $b=7$, $c=-214$. Calculate discriminant: $$\Delta = 7^2 - 4 \times 23 \times (-214) = 49 + 19688 = 19737$$ 11. **Calculate roots:** $$x = \frac{-7 \pm \sqrt{19737}}{46}$$ Approximate $\sqrt{19737} \approx 140.46$. So, $$x_1 = \frac{-7 + 140.46}{46} = \frac{133.46}{46} \approx 2.9$$ $$x_2 = \frac{-7 - 140.46}{46} = \frac{-147.46}{46} \approx -3.2$$ 12. **Check for restrictions:** $x \neq -4$ and $x \neq 3$ (denominators cannot be zero). Both solutions are valid. **Final answer:** $$x \approx 2.9 \text{ or } x \approx -3.2$$