\(-x^3 - 10x^2 + 24x ≤ 0\)

First, get rid of the negative in front of \(x^3\) by multiplying both sides of inequality by -1 and flipping the sign. Then, you will want to factor the inequality:

\(x^3 + 10x^2 - 24x ≥ 0\)
\(x(x^2 + 10x - 24) ≥ 0\)
\(x(x+12)(x-2) ≥ 0\)

The roots of the inequality:

\(x = 0, x = -12, x = 2\)

Plug in numbers to see which intervals are positive.
When x = 3, we get the following: (3)(3+12)(3-2) = (3)(15)(1) > 0.
When x = -1, we get the following: (-1)(-1+12)(-1-2) = (-1)(11)(-3) > 0.

We are done testing numbers and based on the answer choices, C has the correct shaded intervals.