x+2y+(4y^2)/(x-2y)+(4x^2y)/(4y^2-x^2)
= [(x+2y)(x-2y)+4y^2]/(x-2y)+(4x^2y)/(4y^2-x^2)
= x^2/(x-2y)+(4x^2y)/(4y^2-x^2)
= x^2(x+2y)/(x^-4y^2)-4x^2y/(x^2-4y^2)
= (x^3-2x^2y)/(x^2-4y^2)
= x^2(x-2y)/(x+2y)(x-2y)
= x^2/(x+2y)
x+2y+(4y^2)/(x-2y)+(4x^2y)/(4y^2-x^2)
= [(x+2y)(x-2y)+4y^2]/(x-2y)+(4x^2y)/(4y^2-x^2)
= x^2/(x-2y)+(4x^2y)/(4y^2-x^2)
= x^2(x+2y)/(x^-4y^2)-4x^2y/(x^2-4y^2)
= (x^3-2x^2y)/(x^2-4y^2)
= x^2(x-2y)/(x+2y)(x-2y)
= x^2/(x+2y)