$x^2 - 6x - y^2 + 9$
$= (x^2 - 6x +9)-y^2$
$=(x-3)^2 - y^2$
$=(x-y-3)(x+y-3)$
__________
$x^3 + 2x^2 - 6x - 27$
$=(x^3 - 27) + (2x^2 - 6x)$
$=(x-3)(x^2 + 3x + 9) + 2x(x - 3)$
$=(x-3)(x^2 + 3x + 9 + 2x)$
$=(x-3)(x^2 + 5x +9)$
__________
$ x^4 - 4x^3 + 8x^2 -16x + 16$
$=x^4 - 2x^3 - 2x^3 + 4x^2 + 4x^2 - 8x - 8x + 16$
$=x^3(x - 2) - 2x^2(x-2) + 4x(x-2) - 8(x-2)$
$=(x^3 - 2x^2 + 4x - 8)(x-2)$
$=[x^2(x-2) + 4(x-2)](x-2)$
$=(x^2 + 4)(x-2)^2)$