To determine the intervals where the expression is greater than 0, we need to find the zeros of the expression.

First, we find the zeros of each factor:

x^2 - 4 = 0
x^2 = 4
x = ±2

x + 1 = 0
x = -1

x^2 + x + 1 = 0
This quadratic equation does not have real roots. The discriminant is negative, so the factor x^2 + x + 1 is always positive.

Now, we need to consider the signs of the factors in different intervals:
-∞ ... -2: (-)(-)(+) = positive
-2 ... -1: (+)(-)(+) = positive
-1 ... +∞: (+)(+)(+) = positive

Therefore, the expression (x^2 - 4)(x+1)(x^2+x+1) is greater than 0 for x ∈ (-∞, -2) U (-1, +∞).