@JoachimKönig Thank you for the input. I think, requiring, $|x|>1$ solves this trivial answer.

@GerryMyerson I think I got it right this time.

@GerryMyerson Thank you. I did not think it would be this difficult to specify this. wow. Can I just say, trivial answers aside? The suspects being $\{0,1,n\}$ in some combination distributed among $a,b,c,x$.

@DenisShatrov Thank you. I just realized I missed some conditions. I have updated the question appropriately.

@GerryMyerson I mean, sure, but then $f(x)$ is not a degree 2 polynomial....the underlying assumption being that $a\not=0$. The discriminant of a degree $1$ polynomial is not all that interesting.

Other than trying everythng, I am open to anything that is better than the trivial one.

I did get this far. What I am looking for is an algorithm, not a formula, to approximate better and better answers.

@StevenStadnicki, thank you! I will add an edit to the question.

@NoamD.Elkies Sorry, it need NOT be deterministic. G.Melfi the number of "unique" solutions is roughly proportional to the number of factors of $m$, something like that, that is why the condition that $m$ have at least two prime factors. Of course, two odd prime factors....we don't care for $2$ as a factor.

@NoamD.Elkies Integer solutions. The algorithm need to be deterministic, but better than $\mathcal{O}(m^2)$ or even $\mathcal{O}(x_0y_0)$ would be very nice.