Let's simplify the equation step by step:

√x - 2 - √2x + 2 = √2x - 5 - √3x - 1

Combine like terms:

√x - √2x - √3x = 2 - 2 - 5 - 1

Simplify the square roots:

√x(1 - 2 - √3) = -6

Now, we need to isolate the square root using algebraic operations:

√x(-1 - √3) = -6

Divide both sides by (-1 - √3):

√x = -6 / (-1 - √3)

Rationalize the denominator by multiplying by the conjugate of the denominator:

√x = (-6(-1 + √3)) / ((-1 - √3)(-1 + √3))

√x = (6 - 6√3) / (1 - 3)

√x = (6 - 6√3) / -2

√x = -3 + 3√3

Square both sides to solve for x:

x = (-3 + 3√3)^2

x = 9 - 18√3 + 27

x = 36 - 18√3

Therefore, the solution to the equation √x - 2 - √2x + 2 = √2x - 5 - √3x - 1 is x = 36 - 18√3.