To begin solving this inequality, we can first factor each of the quadratic expressions:

(x^2 - 4) = (x+2)(x-2)(x^2 - 4x + 4) = (x-2)(x-2) = (x-2)^2(x^2 - 6x + 8) = (x-2)(x-4)(x^2 + 4x + 4) = (x+2)(x+2) = (x+2)^2

Now, we can rewrite the original inequality as:

((x+2)(x-2))((x-2)^2)((x-2)(x-4))((x+2)^2) >= 0

Simplifying, we get:

(x+2)^3 (x-2)^3 (x-4) >= 0

Since the expression is already factored, we can see that the critical points are x=-2, x=2, and x=4. We can use these critical points to test the sign of the expression in the intervals:

x < -2-2 < x < 22 < x < 4x > 4

We evaluate the sign of the expression in each interval by picking a test point and plugging it into the expression:

Test x=-3: (-1)^3 (-5)^3 (-7) = -875 < 0Test x=0: (2)^3 (-2)^3 (-2) = 32 > 0Test x=3: (5)^3 (1)^3 (1) = 125 > 0Test x=5: (7)^3 (3)^3 (1) = 147 > 0

Therefore, the inequality: (x^2 - 4)(x^2 - 4x + 4)(x^2 - 6x + 8)(x^2 + 4x + 4) >= 0 is satisfied for x ∈ (-2,2] ∪ [4, ∞).