1) \(\left(x+1\right)^3-\left(x-4\right)\left(x+4\right)-x^3\)
\(=x^3+3x^2+3x+1-x^2+16-x^3\)
\(=2x^2+3x+17\)
2) \(\left(x+2\right)^3-x\left(x+3\right)\left(x-3\right)-12x^2-8\)
\(=x^3+6x^2+12x-x\left(x^2-9\right)-12x^2-8\)
\(=x^3-6x^2+12x-x^3+9x-8\)
\(=x^3-6x^2+21x-8\)
3) \(\left(x-2\right)^3-x\left(x-2\right)\left(x+2\right)+6x\left(x-3\right)\)
\(=x^3-6x^2+12x-8-x\left(x^2-4\right)+6x^2-18x\)
\(=x^3-6x-8-x^3+4x\)
\(=-2x-8\)
4) \(x\left(x-5\right)\left(x+5\right)-\left(x-5\right)^3+100x\)
\(=x\left(x^2-25\right)-\left(x^3-15x^2+75x-125\right)+100x\)
\(=x^3-25x-x^3+15x^2-75x+125+100x\)
\(=15x^2+100\)
5) \(\left(x-3y\right)^3-\left(x-2y\right)\left(2y+x\right)\)
\(=x^3-9x^2y+27xy^2-27y^3-\left(x^3-8y^3\right)\)
\(=x^3-9x^2y+27xy^2-27y^3-x^3+8y^3\)
\(=-9x^2y+27xy^2-19y^3\)
6) \(\left(-x-2y\right)^3+x\left(2y-x\right)\left(x+2y\right)\)
\(=-x^3-6x^2y-12xy^2-8y^3+x\left(4y^2-x^2\right)\)
\(=-x^3-6x^2y-12xy^2-8y^3+4xy^2-x^3\)
\(=-2x^3-6x^2y-8xy^2-8y^3\)
7) \(-\left(2x-y\right)^3-x\left(2x-y\right)^2-y^3\)
\(=-\left(8x^3-12x^2y+6xy^2-y^3\right)-x\left(4x^2-4xy+y^2\right)-y^3\)
\(=-8x^3+12x^2y-6xy^2+y^3-4x^3+4x^2y-xy^2-y^3\)
\(=-12x^3+16x^2y-7xy^2\)
8) \(-x\left(x-y\right)^2+\left(x-y\right)^3+y^2\left(y-2x\right)\)
\(=-x\left(x^2-2xy+y^2\right)+x^3-3x^2y+3xy^2-y^3+y^3-2xy^2\)
\(=-x^3+2x^2y-xy^2+x^3-3x^2y+3xy^2-y^3+y^3-2xy^2\)
\(=-x^2y\)
