1. **State the problem:** Solve the equation $$\frac{4}{x} - \frac{5}{x+2} = \frac{1}{24}$$ for $x$. 2. **Identify the common denominator:** The denominators are $x$, $x+2$, and $24$. The least common denominator (LCD) is $$24x(x+2).$$ 3. **Multiply both sides by the LCD to clear fractions:** $$24x(x+2) \times \left(\frac{4}{x} - \frac{5}{x+2}\right) = 24x(x+2) \times \frac{1}{24}$$ 4. **Simplify each term:** $$24(x+2) \times 4 - 24x \times 5 = x(x+2)$$ 5. **Write the expanded form:** $$96(x+2) - 120x = x^2 + 2x$$ 6. **Distribute and simplify:** $$96x + 192 - 120x = x^2 + 2x$$ 7. **Combine like terms on the left:** $$-24x + 192 = x^2 + 2x$$ 8. **Bring all terms to one side:** $$0 = x^2 + 2x + 24x - 192$$ 9. **Simplify:** $$0 = x^2 + 26x - 192$$ 10. **Solve the quadratic equation:** Use the quadratic formula $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ where $a=1$, $b=26$, and $c=-192$. 11. **Calculate the discriminant:** $$\Delta = 26^2 - 4 \times 1 \times (-192) = 676 + 768 = 1444$$ 12. **Find the square root:** $$\sqrt{1444} = 38$$ 13. **Calculate the roots:** $$x = \frac{-26 \pm 38}{2}$$ 14. **First root:** $$x = \frac{-26 + 38}{2} = \frac{12}{2} = 6$$ 15. **Second root:** $$x = \frac{-26 - 38}{2} = \frac{-64}{2} = -32$$ 16. **Check for restrictions:** The denominators $x$ and $x+2$ cannot be zero, so $x \neq 0$ and $x \neq -2$. Both $6$ and $-32$ are valid. **Final answer:** $$x = 6 \text{ or } x = -32$$