\(\left(a-x-y\right)^3-\left(a+x-y\right)^3\)
\(=\left(a-x-y-a-x+y\right)\left[\left(a-x-y\right)^2+\left(a-x-y\right)\left(a+x-y\right)+\left(a+x-y\right)^2\right]\)
\(=-2x\left[\left(a-x-y\right)^2+\left(a+x-y\right)^2+\left(a-y\right)^2-x^2\right]\)
\(=-2x\left[\left(a-y\right)^2-2x\left(a-y\right)+x^2+\left(a-y\right)^2+2x\left(a-y\right)+x^2+\left(a-y\right)^2-x^2\right]\)
\(=-2x\left[3\left(a-y\right)^2+x^2\right]\)