To simplify the expression, let's first expand the square root term:

√(x^2-4xy +4y^2)^2 = √((x-2y)^2)^2 = √(x-2y)^2

Now, since x ≥ 2y, we can rewrite x as 2y + k, where k is a non-negative number or zero.

√(x-2y)^2 = √((2y + k) - 2y)^2 = √k^2 = |k|

Since k is non-negative or zero, |k| = k.

Therefore, the simplified expression is:

x + 2y - √(x^2-4xy +4y^2)^2 = 2y + k + 2y - k = 4y