Suppose I am given a finite set of rational numbers and I am told that these are fractions of the form $(an^2+bn+c)/(dn+e)$ for fixed integers $a,b,c,d,e$ and varying integer $n$. I am not told what $a,b,c,d,e$ and the different $n$ are. How am I supposed to find $a,b,c,d,e$ if I have sufficiently large and general amount of numbers in my set. This is not relevant to me, but out of curiosity, what is the minimum cardinality I need for this (general) set to be able to uniquely determine $a,b,c,d$ and $e$.
I was thinking of taking the least common multiple of the difference of the denominators of the fractions to get $e$, but I am not guaranteed that the denominators I get are exactly $dn+e$ (they could be a factor of it, since the numerator and denominator of $(an^2+bn+c)/(dn+e)$ need not be coprime.