a. $81x^2 - 6yz- 9y^2 - z^2 {}=81x^2-(9y^2+6yz+z^2)=81x^2-(3y+z)^2=(9x)^2-(3y+z)^2=(9x-3y-z)(9x+3y+z)$
b. $ xz-yz-x^2+2xy-y^2 {}=(zx-yz)-(x^2-2xy+y^2)=z(x-y)-(x-y)^2=(x-y)(z-x+y)$
c. $x^{2}+8x+15=x^2+3x+5x+15=x(x+3)+5(x+3)=(x+3)(x+5)$
d. $x^2-x-12=x^2-x-16+4=(x^2-16)-(x-4)=(x-4)(x+4)-(x-4)=(x-4)(x+4-1)=(x-4)(x+3)$
e. $81x^4+4^{}=81x^4+36x^2+4-36x^2=(9x^2+2)^2-(6x)^2=(9x^2-6x+2)(9x^2+6x+2)$