How to sort this two elements of a list?

Question

Hi So I have the following list:

data = {{3.73324*10^10 + 147881. s + 1. s^2, 1}, {3.73324*10^10 + 357017. s + 1. s^2, 1}}

(this is a short example, in this case the list has only two elements, but this is part of a big scheme of things)

I want to order the elements of this list based on a parameter Q

Q1 = Sqrt[CoefficientList[data[[1]][[1]], s][[1]]]/CoefficientList[data[[1]][[1]], s][[2]]
Q2 = Sqrt[CoefficientList[data[[2]][[1]], s][[1]]]/CoefficientList[data[[2]][[1]], s][[2]]

I want to order it by increasing values of Q so, in this case the elements should be swapped. What is the best way to do this in systematic way using sort? I have tried this but failed miserably, probably due to syntax:

Sort[FactorsDen, Sqrt[CoefficientList[#1[[2]][[1]], s][[1]]]/CoefficientList[#1[[2]][[1]], s][[2]]< Sqrt[CoefficientList[#2[[2]][[1]], s][[1]]]/CoefficientList[#2[[2]][[1]], s][[2]] &]

Any guess on how can I do this in an easy way? Thank you!

Answer 1

$Version

(* "13.2.1 for Mac OS X ARM (64-bit) (January 27, 2023)" *)

Clear["Global`*"]

metric[x_List] := 
 Module[{t = CoefficientList[x[[1]], s]}, Sqrt[t[[1]]]/t[[2]]]

data = {{3.73324*10^10 + 147881. s + 1. s^2, 
    1}, {3.73324*10^10 + 357017. s + 1. s^2, 1}};

SortBy[data, metric]

(* {{3.73324*10^10 + 357017. s + 1. s^2, 
  1}, {3.73324*10^10 + 147881. s + 1. s^2, 1}} *)