To solve the inequality ((x^2 - 2)/(x + 1))^2 > 0, we need to consider the different intervals where the expression can be greater than zero.

First, let's find the critical points by setting the expression equal to zero:

(x^2 - 2)/(x + 1) = 0
x^2 - 2 = 0
x^2 = 2
x = ±√2

Therefore, the critical points are x = -√2 and x = √2. We can use these values to create our intervals.

Interval 1: x < -√2
Choose x = -3:
((-3^2 - 2)/(-3 + 1))^2 = (7/-2)^2 = (-3.5)^2 = 12.25 > 0

Interval 2: -√2 < x < √2
Choose x = 0:
((0^2 - 2)/(0 + 1))^2 = (-2/1)^2 = (-2)^2 = 4 > 0

Interval 3: x > √2
Choose x = 3:
((3^2 - 2)/(3 + 1))^2 = (7/4)^2 = (1.75)^2 = 3.0625 > 0

Therefore, the solution to the inequality ((x^2 - 2)/(x + 1))^2 > 0 is x < -√2 or x > √2.