`1)`
`x^2+y^2+2xy+2x+2y+1`
`=(x^2+2xy+y^2)+(2x+2y)+1`
`=(x+y)^2+2(x+y)+1`
`=(x+y+1)^2`
`2)`
`x^6+27`
`=(x^2+3)(x^4-3x^2+9)`
`=(x^2+3)(x^4-3x^3+3x^3+3x^2+3x^2-9x^2+9x-9x+9`
`=(x^2+3)[(x^4-3x^3+3x^2)+(3x^3-9x^2+9x)+(3x^2-9x+9)]`
`=(x^2+3)[x^2(x^2-3x+3)+3x(x^2-3x+3)+3(x^2-3x+3)]`
`=(x^2+3)(x^2-3x+3)(x^2+3x+3)`
`3)`
`x^3+x^2y-x^2z-xyz`
`=x(x^2+xy-xz-yz)`
`=x[(x^2+xy)-(xz+yz)]`
`=x[x(x+y)-z(x+y)]`
`=x(x+y)(x-z)`
`4)`
`x^2-xy-5x+5y`
`=(x^2-xy)-(5x-5y)`
`=x(x-y)-5(x-y)`
`=(x-y)(x-5)`
`5)`
`x^2+(2x+y)y-z^2`
`=x^2+2xy+y^2-z^2`
`=(x^2+2xy+y^2)-z^2`
`=(x+y)^2-z^2`
`=(x+y-z)(x+y+z)`
`6)`
`x^2-5x-14`
`=x^2+2x-7x-14`
`=x(x+2)-7(x+2)`
`=(x+2)(x-7)`