Answer
The Laplace's equation is satisfied.
Work Step by Step
We need to verify Laplace's equation
$\nabla^2 f =\dfrac{\partial^2 f}{\partial x^2}+\dfrac{\partial^2 f}{\partial y^2}+\dfrac{\partial^2 f}{\partial z^2}$
We take the first partial derivative of the given function $f(x,y,z)$ with respect to $x$, by treating $y$ and $z$ as a constant, and vice versa:
$f_x=-6xz \\ f_y=-6yz \\f_z=6z^2-3x^2-3y^2$
and $\nabla^2 f =-6z-6z+12z=0$
So, the Laplace's equation is satisfied.