How to factor polynomials in $\mathbb{Z}_{n}[x]$ I realize there already is a question almost identical to this (here), but the answers given are a bit too vague.
The problem I have is that I'm looking to factor the polynomial $$x^2+23x+18 \text{ 
in } \mathbb{Z}_{28}$$ into as many ways as possible. I've tried through trial and error since we're looking for $a,b \in \mathbb{Z_{28}}$ such that $$(x+a)(x+b)=x^2+23x+18$$ So $ab=18$ and $a+b=23$. But this seems too tedious and I haven't found an answer yet. So if anyone has a method for me to follow (sort of a recipe) or some clues as to how I can discover the method myself it would be much appreciated.
Thanks.

When there are zero divisor factoring a polynomial becomes very hairy. You have to define the terms very carefully to get much done. I suppose you can have fun with this, but I list a few things you need to be aware of:


*

*Are linear polynomials irreducible? Modulo $28$ we have "factorizations" like
$$(7x-1)(8x+1)=-x-1.$$ The leading coefficients in the left are necessarily zero divisors, but?

*When the modulus is not square-free we have non-trivial units in the ring. Here
$$(14x+1)^2=1.$$


For more information, check out: 


*

*The sources Bill Dubuque dug
out.

*More local discussion - not sure if they get to the bottom of it.

