To solve this equation, we first notice that we can rewrite the absolute value expression as two separate cases: when x^2 + x is greater than or equal to 0, and when x^2 + x is less than 0.

Case 1: x^2 + x >= 0
In this case, the absolute value expression simplifies to x^2 + x. So the equation becomes (x^2 + x)^2 + (x^2 + x) - 2 = 0.

Expanding the left side:
(x^2 + x)^2 = x^4 + 2x^3 + x^2
(x^2 + x) = x^2 + x

Adding these together:
(x^4 + 2x^3 + x^2) + (x^2 + x) - 2 = 0
x^4 + 2x^3 + 2x^2 + x - 2 = 0

This is a quartic equation that can be difficult to solve without the use of a calculator or computer algebra system.

Case 2: x^2 + x < 0
In this case, the absolute value expression simplifies to -(x^2 + x). So the equation becomes (x^2 + x)^2 - (x^2 + x) - 2 = 0.

Expanding the left side:
(x^2 + x)^2 = x^4 + 2x^3 + x^2
-(x^2 + x) = -x^2 - x

Adding these together:
(x^4 + 2x^3 + x^2) - (x^2 + x) - 2 = 0
x^4 + 2x^3 + 2x^2 - x - 2 = 0

This is also a quartic equation that can be difficult to solve without a calculator or computer algebra system.

In general, solving quartic equations can be quite challenging and often requires numerical methods or other advanced techniques.