Which of the following shaded intervals is the graph of the inequality

I created this question to demonstrate the difference between high school math and GMAT math.

Students who insist on performing high school math for every question will undoubtedly factor the expression to get something like \(-x(x+12)(x-2) ≤ 0\) or perhaps \(x(x+12)(x-2) ≥ 0\).
Then they'll test a variety of values (or apply some number sense) to correctly identify the intervals that satisfy the inequality.
That's the approach that would definitely please your high school math teacher .

Alternatively, since our goal is to identify the correct answer AND do so in an expedient manner, we can test some values of x to determine whether they satisfy the given inequality. In my opinion, students are less likely to make mistakes while testing values. Plus, it's very fast. So let's go with that approach....

When I scan the answer choices (always scan the answer choices before committing to a specific approach), I see that some answers show that x = -1 is a solution, and others show that x = -1 is not a solution.

So let's test this by plugging x = -1 into the given inequality to get: \(-[(-1)^3] - 10(-1)^2 + 24(-1) ≤ 0\)
Simplify: \(-33 ≤ 0\), which is TRUE, which means x = -1 is a solution.
This means we can eliminate answer choices A, D and E, because they show that x = -1 is NOT a solution.

With that super easy/fast step, we're down answer choices B and C

Now notice that choice C suggests that x = 3 is a solution to the inequality, and choice B suggests that x = 3 is NOT a solution.
When we test x = 3, we get: \(-(3^3) - 10(3)^2 + 24(3) ≤ 0\)
Simplify: \(-45 ≤ 0\), which is TRUE, which means x = 3 is a solution.
This means we can eliminate answer choice B, because it shows that x = 3 is NOT a solution.

By the process of elimination, the correct answer is C