Answer
Not homogeneous
Work Step by Step
Recall that the definition of a homogeneous function is a function that satisfies the following condition
$f(tx,ty)=t^nf(x,y)$ where $n$ is the degree.
If $f(x,y)=x^3+3x^2y^2-2y^2$
all we have to do is plug in the point $(tx,ty)$
$f(tx,ty)=(tx)^3+3(tx)^2(ty)^2-2(ty)^2$
$=t^3x^3+3t^4x^2y^2-2t^2y^2$
Right away we can see that this function is not homogeneous as there are different powers of $t$ in each term.
