We have,

$9 x^{2}+4 y^{2}+16 z^{2}+12 x y-16 y z-24 x z$

$=(3 x)^{2}+(2 y)^{2}+(-4 z)^{2}+2(3 x)(2 y)+2(2 y)(-4 z)+2(-4 z)(3 x)$

$=\{3 x+2 y+(-4 z)\}^{2}\left[\because a^{2}+b^{2}+c^{2}+2 a b+2 b c+2 c a=(a+b+c)^{2}\right]$

$=(3 x+2 y-4 z)^{2}=(3 x+2 y-4 z)(3 x+2 y-4 z)$