Can someone please check this to see if I have reduced to simplest terms:

{(-x-y)X(x-y)}/{(x-y)^2 + (x+y)^2}

I get -1/2 (x-y)(x+y)divided by (x^2+y^2).
Is this right and if so, can it be simplified any more? Thanks so much.

To check if the expression (-x-y)(x-y)/[(x-y)^2 + (x+y)^2] is simplified to its simplest terms, we will simplify the expression and see if any further simplification is possible.

First, let's simplify the numerator and denominator separately:

Numerator:
(-x-y)(x-y) = -x(x-y) - y(x-y)
= -x^2 + xy - yx + y^2
= -x^2 + y^2

Denominator:
(x-y)^2 + (x+y)^2 = x^2 - 2xy + y^2 + x^2 + 2xy + y^2
= 2x^2 + 2y^2

Now, let's substitute the simplified numerator and denominator back into the original expression:

(-x^2 + y^2)/(2x^2 + 2y^2)

This expression cannot be simplified further because there are no common factors between the numerator and the denominator to cancel out.

Therefore, your simplified expression (-x^2 + y^2)/(2x^2 + 2y^2) is correct and cannot be simplified any further.