Let's simplify the equation step by step:

Expand the squared term: (x^2 - 2x + 2)^2 = (x^2 - 2x + 2)(x^2 - 2x + 2)

= x^4 - 2x^3 + 2x^3 - 4x^2 + 4x^2 - 4x + 4x + 4

= x^4 - 4x^3 + 4x^2 + 0x + 4

Now place the expanded term back into the equation:

x^4 - 4x^3 + 4x^2 + 0 - x^2 + 2x - 22 = 0

Combine like terms:

x^4 - 4x^3 + 3x^2 + 2x - 22 = 0

Therefore, the simplified form of the equation (x^2 - 2x + 2)^2 - x^2 + 2x - 22 = 0 is x^4 - 4x^3 + 3x^2 + 2x - 22 = 0.