I understand the Stars and Bars (balls in bins) method that is commonly used to tackle questions like "Find the number of nonnegative integer solutions to the equation $x_1+x_2+x_3 +x_4= n$." and various extensions of this problem, but I am curious if this can be applied for questions like:
$$(x_1+1)(x_2+1)(x_3+1)(x_4+1) = (9-x_1)(5-x_2)(3-x_3)(2-x_4).$$
where all factors on the right hand side are nonnegative integers. The solution to this question is $6$, where $(x_1,x_2,x_3,x_4)$ is either $(4,1,1,1)$, $(3,2,2,0)$, $(1,3,1,1)$, $(7,1,1,0)$, $(5,2,0,1)$, or $(4,3,1,0)$. How can I approach this question combinatorically? Can this similar approach be extended for $n$ variables? Thank you!