Final answer:
The solution to the inequality x^4 + 4x^3 ≤ 12x^2 can be found by factoring and using the interval test, yielding the solution set -6 ≤ x ≤ 2.
Step-by-step explanation:
To solve the inequality x4 + 4x3 ≤ 12x2, first we can factor out x2 from the left side of the inequality:
x2(x2 + 4x - 12) ≤ 0
Now, we factor the quadratic equation inside the parentheses:
(x2 + 6x)(x2 - 2x) ≤ 0
Setting each factor equal to zero, we find where the function touches or crosses the x-axis (x-intercepts):
These values divide the number line into four intervals. We need to determine the sign of the function in each interval to find out the solution set. Applying the interval test, we can see that the function is non-positive (less than or equal to zero) between x = -6 and x = 0, and between x = 0 and x = 2.
Therefore, the solution to the inequality is:
-6 ≤ x ≤ 2