Counting reducible monic polynomials in $F_p[x]$

Question

this question is related to the ones here and here. I am working on the same problem ("how many monic reducible quadratic polynomials are there in $F_p[x]$"), and I'm unsure about the counting argument: there are $p$ polynomials of the form $(x-a)^2$ - ok - but why are there $\frac{p(p-1)}{2}$ of the form $(x-a)(x-b)$? How does this deal with $ab \equiv bc \mod p$ for different $a,b$? Thanks.