We have,

$(2 x-y+3 z)\left(4 x^{2}+y^{2}+9 z^{2}+2 x y+3 y z-6 x z\right)$

$=\{2 x+(-y)+3 z\}\left\{(2 x)^{2}+(-y)^{2}+(3 z)^{2}-(2 x)(-y)-(-y)(3 z)-(3 z)(2 x)\right\}$

$=(2 x)^{3}+(-y)^{3}+(3 z)^{3}-3(2 x)(-y)(3 z)$ $\left[\because(a+b+c)\left(a^{2}+b^{2}+c^{2}-a b-b c-c a\right)=a^{3}+b^{3}+c^{3}-3 a b c\right]$

$=8 x^{3}-y^{3}+27 z^{2}+18 x y z$