The equation $3x^2-2y^2-4z^2+xy+xz+6yz=0$ represents

The given equation is $3x^2-2y^2-4z^2+xy+xz+6yz=0$

Consider

$\begin{array}{rcl}3x^2+3xy-2xy-2y^2& =& 3x\left(x+y\right)-2y\left(x+y\right)\\ & =& \left(3x-2y\right)\left(x+y\right)\end{array}$

Hence, two factors of the given equation are $\left(3x-2y+2z\right)\left(x+y-2z\right)$

Hence, the equation $3x^2-2y^2-4z^2+xy+xz+6yz=0$ can be written as $\left(3x-2y+2z\right)\left(x+y-2z\right)=0$

These factors are linear equations in terms of $x,y,z$ and no constant terms.

Therefore, the given equation represents a pair of planes which are passing through the origin