\(x^2+y^2-x^2y^2+xy-x-y\)
\(=x^2-x^2y^2+y^2-y+xy-x\)
\(=x^2\left(1-y^2\right)+y\cdot\left(y-1\right)+x\left(y-1\right)\)
\(=-x^2\left(y-1\right)\left(y+1\right)+\left(y-1\right)\left(x+y\right)\)
\(=\left(y-1\right)\left(-x^2y-x^2+x+y\right)\)
\(=\left(y-1\right)\left[y\left(-x^2+1\right)-x\left(x-1\right)\right]\)
\(=\left(y-1\right)\left[-y\left(x-1\right)\left(x+1\right)-x\left(x-1\right)\right]\)
\(=\left(y-1\right)\left(x-1\right)\left(-xy-y-x\right)\)
\(x^2+y^2+2xy+2x+2y+1\)
\(=\left(x+y\right)^2+2\left(x+y\right)+1\)
\(=\left(x+y+1\right)^2\)
\(\left(x-y\right)^2+2\left(y-x\right)+1\)
\(=\left(x-y\right)^2-2\left(x-y\right)+1^2\)
\(=\left(x-y-1\right)^2\)
\(x^3-4x^2-8x+8\)
\(=\left(x^3+8\right)-4x\left(x+2\right)\)
\(=\left(x+2\right)\left(x^2-2x+4\right)-4x\left(x+2\right)\)
\(=\left(x+2\right)\left(x^2-6x+4\right)\)
`1)x^2 + y^2 - x^2.y^2 + xy - x - y`
`= x^2 - x^2.y^2 + xy - x +y^2 - y`
`=x^2(1 - y^2) + x(y - 1) +y(y - 1)`
`= x^2(1-y)(1+y) + x(y-1) + y(y-1)`
`=-x^2(y-1)(1+y) + x(y-1) + y(y-1)`
`=(y-1)[-x^2(1+y)+x + y]`
`=(y-1)(-x^2-x^2y + x + y)`
`=(y-1)[(-x^2+x)-(x^2y-y)]`
`=(y-1)[-x(x-1) - y(x^2 - 1)]`
`=(y-1)[(-x(x-1) -y(x-1)(x+1)]`
`=(y-1)[(x-1)(-x-y(x+1)]`
`=(y-1)(x-1)(-x-xy-y)`
`2)x^2 + y^2 +2xy + 2x + 2y + 1`
`=(x^2 + 2xy + y^2) + 2(x+y) + 1`
`= (x+ y)^2 + 2(x + y) + 1`
`=(x + y)^2 + 2(x + y) . 1 + 1^2`
`= (x + y + 1)^2`
`3)(x-y)^2 + 2(y-x) + 1`
`=(y-x)^2 + 2(y-x) + 1`
`=(y-x)^2 + 2(y-x).1 + 1^2`
`=(y-x + 1)^2`
`4)x^3 - 4x^2 - 8x + 8`
`=(x^3 + 8) - (4x^2 + 8x)`
`=(x^2 + 2^3) - 4x(x + 2)`
`= (x+2)(x^2 - 2x + 4) - 4x(x+2)`
`=(x+2)(x^2 - 2x + 4 - 4x)`
`=(x+2)(x^2-6x + 4)`
