Consider the graph in 3 -space of \[ 0=x y z(x+y)(y+z)(z+x)(x-y)(y-z)(z-x) \] This graph divides 3 -space into $N$ connected regions. What is $N$ ?
Note that reflecting for each choice of sign for $x, y, z$, we get new regions. Therefore, we can restrict to the case where $x, y, z>0$. In this case, the sign of the expression only depends on $(x-y)(y-z)(z-x)$. It is easy to see that for this expression, every one of the $3 !=6$ orderings for $\{x, y, z\}$ contributes a region. Therefore, our answer is $2^{3} \cdot 3 !=48$. $\fbox{48}$.