Simplify each of the following expressions into simplest terms.

a) {[-X+Y+(-2)*(-2Y)*[X-Y]}/[(X-Y)^2+(X+Y)^2]

b)
(X^3+4X^2Y+4XY^2+2XY-6Y^2)/(X+2Y)

4 answers

Please don't capitalize your variables, it makes it hard to read them.

(-x + y - 2(-2y)(x - y) )/( (x-y)^2 + (x+y)^2 )
= (-x + y + 4xy - 4y^2)/(x^2 - 2xy + y^2 + x^2 + 2xy + y^2)
= ( x(4y - 1) - y(4x - 1) )/( 2(x^2 + y^2) )
= (4x-1)(x-y) / (2(x^2 + y^2) )

for the second, I don't see much happening
you could separate the fraction into

(x^3 + 4x^2y + 4xy^2)/(x+2y) + (2xy - 6y^2)/(x+2y)
= x(x^2 + 4xy + 4y^2)/(x+2y) + 2y(x - 3y)/(x-2y)
= x(x-2y)^2 / (x-2y) + 2y(x-3y)/(x-2y
= x(x-2y) + 2y(x-3y)/(x-2y)

if that second-last term had been 3xy we could have done more.
Was there a typo ?
no no typo the equation is
(x^3+4x^2y+4xy^2+2xy-6y^2)/(x+2y)

sorry about the caps on my first post
i did however seem to have made a typo on the first equation the right one is

{[-x+y+(-2y)]*[x-y]}/[(x-y)^2+(x+y)^2]
If no typo in the second, then my answer stands as is

for the first, follow my steps with your corrected expression. There should be only minor changes.