To simplify this expression, we can first let (y = (x+2)^2). Then the expression becomes:

(y^2 - 5y + 4 = 0)

Now, we can factor this quadratic equation:

((y-4)(y-1) = 0)

So, (y = 4) or (y = 1). Since (y = (x+2)^2), we can substitute back to get:

((x+2)^2 = 4) or ((x+2)^2 = 1)

This gives us two possible solutions:

(x+2 = 2) or (x+2 = -2)

Therefore, the solutions to the equation ((x+2)^4 - 5(x+2)^2 + 4 = 0) are (x = 0) and (x = -4).