x+2y+4y^2/x-2y+4x^2y/4y^2-x^2=

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x+2y+4y^2/x-2y+4x^2y/4y^2-x^2=x+2y+4y^2/x-2y+4x^2y/4y^2-x^2=x+2y+4y^2/x-2y+4x^2y/4y^2-x^2=x+2y+4y^2/

x+2y+4y^2/x-2y+4x^2y/4y^2-x^2=
x+2y+4y^2/x-2y+4x^2y/4y^2-x^2=

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