1. **State the problem:** Solve the equation $$\frac{5}{24 - 3x} = -6x$$. 2. **Formula and rules:** To solve for $x$, multiply both sides by the denominator to clear the fraction, then solve the resulting equation. 3. **Multiply both sides by $24 - 3x$:** $$5 = -6x(24 - 3x)$$ 4. **Distribute $-6x$:** $$5 = -144x + 18x^2$$ 5. **Rearrange to standard quadratic form:** $$18x^2 - 144x - 5 = 0$$ 6. **Simplify by dividing all terms by the greatest common factor 1 (no change here):** $$18x^2 - 144x - 5 = 0$$ 7. **Use the quadratic formula:** $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ where $a=18$, $b=-144$, $c=-5$. 8. **Calculate the discriminant:** $$b^2 - 4ac = (-144)^2 - 4(18)(-5) = 20736 + 360 = 21096$$ 9. **Calculate the square root:** $$\sqrt{21096} \approx 145.26$$ 10. **Calculate the two solutions:** $$x = \frac{144 \pm 145.26}{36}$$ 11. **First solution:** $$x_1 = \frac{144 + 145.26}{36} = \frac{289.26}{36} \approx 8.035$$ 12. **Second solution:** $$x_2 = \frac{144 - 145.26}{36} = \frac{-1.26}{36} \approx -0.035$$ 13. **Check for restrictions:** The denominator $24 - 3x \neq 0$ so $x \neq 8$. 14. **Conclusion:** The solution $x \approx 8.035$ is invalid because it makes the denominator close to zero and is outside the domain. The valid solution is: $$\boxed{x \approx -0.035}$$