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 ?