To solve this inequality, we need to consider two cases: when the expression inside the absolute value on the left side is positive and when it is negative.
Case 1: (x^3-2x^2+12 \geq 0) This simplifies to (x^3-2x^2+12 \leq x^3+2x^2+4). Simplifying further, we have (-4x^2+8 \leq 4). (x^2 \geq 3). This inequality holds true for all real values of x.
Case 2: (x^3-2x^2+12 < 0) This simplifies to (-(x^3-2x^2+12) \leq x^3+2x^2+4). Simplifying further, we have (x^2-4 \leq 4). (x^2 \leq 8). This inequality holds true for (x \in [-\sqrt{8}, \sqrt{8}]).
Therefore, the solution to the inequality (|x^3-2x^2+12| \leq |x^3+2x^2+4|) is (x \in [-\sqrt{8}, \sqrt{8}]).
To solve this inequality, we need to consider two cases: when the expression inside the absolute value on the left side is positive and when it is negative.
Case 1: (x^3-2x^2+12 \geq 0)
This simplifies to (x^3-2x^2+12 \leq x^3+2x^2+4).
Simplifying further, we have (-4x^2+8 \leq 4).
(x^2 \geq 3).
This inequality holds true for all real values of x.
Case 2: (x^3-2x^2+12 < 0)
This simplifies to (-(x^3-2x^2+12) \leq x^3+2x^2+4).
Simplifying further, we have (x^2-4 \leq 4).
(x^2 \leq 8).
This inequality holds true for (x \in [-\sqrt{8}, \sqrt{8}]).
Therefore, the solution to the inequality (|x^3-2x^2+12| \leq |x^3+2x^2+4|) is (x \in [-\sqrt{8}, \sqrt{8}]).