Determine the number of reducible degree two polynomials in $Z_5[x]$.
My toolbox only consists of Gauss Theorem and Eisentsein's Criterion alongside all the other basic rules of Ring Theory. So a degree two polynomial will be reducible if it can be written as the product as two linear polynomials. So it would suffice to find the roots of an arbitrary degree two polynomial in $Z_5[x]$. The part I am struggling with here is the fact that the polynomial is not necessarily monic. So I cannot simply write $p(x) = (x-a)(x-b)$, in which case it would simply be ${5 \choose 2}=10$. Plus we are working in $Z_5[x]$, so an arbitrary coefficient $a_n$ will not be a unit, so I cannot make it monic. I am stuck at this point and any advice is appreciated. Thanks!