2x^2-12xy+18y^2+8xz-24yz+6z^2

2x^2-12xy+18y^2+8xz-24yz+6z^2
=2(x^2-6xy+9y^2+4xz-12yz+3z^2)

x^2-6xy+9y^2+4xz-12yz+3z^2
=(x+ay+bz)(a+cy+dz)
=x^2+(a+c)y+acy^2+(b+d)xz+(ad+bc)yz+bdz^2
a+c=-6
ac=9
b+d=4
ad+bc=-12
bd=3

a+c=-6
ac=9
a=-6-c
-6c-c^2=9
c^2+6c+9=0
(c+3)^2=0
c=-3,a=-3

b+d=4,b=4-d
bd=3
4d-d^2=3
d^2-4d+3=0
(d-1)(d-3)=0
d=1,b=3
d=3,b=1

符合ad+bc=-12

所以=2(x-3y+z)(x-3y+3z)

设2x^2-12xy+18y^2+8xz-24yz+6z^2
=(ax+by+cz)(dx+ey+fz)
解得
2x^2-12xy+18y^2+8xz-24yz+6z^2
=2(x - 3 y + z)(x - 3 y + 3 z)