Let 
$$
f_{a,b}(x)=x^3+(u-1-a-b)x^2+ax+b,
$$ 
where $u\in\mathbb{Z}_p^*$ is fixed. Let $S$ be the set consisting of all pairs $(a,b)\in\mathbb{Z}_p^2$ such that $f_{a,b}(x)$ factor linearly. Then what is the cardinality of $S$?
Is it possible to get an exact formula somehow ?