To solve the equation, we can set each factor equal to zero and solve for x.

First, let's set |x| + 2 equal to zero:
|x| + 2 = 0

To eliminate the absolute value, we can consider two cases:

Case 1: x is positive or zero.
If x is positive or zero, then |x| = x. So we have:
x + 2 = 0
x = -2

Case 2: x is negative.
If x is negative, then |x| = -x. So we have:
-x + 2 = 0
x = 2

Therefore, the solutions for |x| + 2 = 0 are x = -2 and x = 2.

Next, let's set 3x² - 12 equal to zero:
3x² - 12 = 0

We can factor out a common factor of 3:
3(x² - 4) = 0

Now, we can set each factor equal to zero:
x² - 4 = 0

This is a difference of squares, so we can factor it further:
(x - 2)(x + 2) = 0

Setting each factor equal to zero, we have:
x - 2 = 0 or x + 2 = 0

Solving for x in each equation, we get:
x = 2 or x = -2

Therefore, the solutions for 3x² - 12 = 0 are x = 2 and x = -2.

In conclusion, the solutions for the equation (|x| + 2)(3x² - 12) = 0 are x = -2, x = 2.