`1) x^2 - x - y^2 - y `

`= (x^2 - x) - (y^2 + y) `

`= x(x - 1) - (y^2 + y) `

`= x(x - 1) - y(y + 1)`

`2) x^2 - 2xy + y^2 - z^2 `

`= (x - y)^2 - z^2 `

`= [(x - y) - z][(x - y) + z]`

`3) 5x - 5y + ax - ay`

`= (5x + ax) - (5y + ay) `

`= x(5 + a) - y(5 + a)`

`= (5 + a)(x - y) `

`4) a^3 - a^2x - ay + xy`

`= (a^3 - a^2x) - (ay - xy) `

`= a^2(a - x) - y(a - x)`

`= (a - x)(a^2 - y) `

`5) x^3 - x + y^3 - y `

`= (x^3 + y^3) - (x + y)`

`= (x + y)(x^2 - xy + y^2) - (x + y) `

`= (x + y)[x^2 - xy + y^2 - 1]`

`6) x^4 + 2x^3 + x^2`

`= x^2(x^2 + 2x + 1) `

`= x^2(x + 1)^2`