To simplify the expression, let's expand both terms step by step:

Expand (x^2 + 2x)^2:
(x^2 + 2x)^2 = (x^2 + 2x)(x^2 + 2x)
= x^4 + 2x^3 + 2x^3 + 4x^2 (using FOIL method for multiplication)
= x^4 + 4x^3 + 4x^2

Expand 2(x + 1)^2:
2(x + 1)^2 = 2(x^2 + 2x + 1)
= 2x^2 + 4x + 2

Now, we can substitute these expanded forms back into the original expression:

(x^2 + 2x)^2 - 2(x + 1)^2 = (x^4 + 4x^3 + 4x^2) - 2(2x^2 + 4x + 2)
= x^4 + 4x^3 + 4x^2 - 4x^2 - 8x - 4
= x^4 + 4x^3 - 8x - 4

Therefore, (x^2 + 2x)^2 - 2(x + 1)^2 simplifies to x^4 + 4x^3 - 8x - 4.