Best way is to apply log to the function as optimal solution to log of a function is the same for the function.
$ \log f = p[\log (C(x+t)+ mx(x+t)+s)-\log(x+t)]$
Then you have to get the actual solution by the exp of $\log$ base used (e, 2 or 10).
  Solvers won't have a problem with 2nd degree $x^2$

Best way is to apply log to the function as optimal solution to log of a function is the same for the function.
$ \log f = p[\log (C(x+t)+ mx(x+t)+s)-\log(x+t)]$
Then you have to get the actual solution by the exp of $\log$ base used (e, 2 or 10).

However log functions need to be transformed into auxillary variables. You can also use the exp with $p$ but these also have to expressed as additional variables using addition constraints. Here is the link to suggested piecewise solution by Gurobi. Gurobi provides constraints as model methods like addGenConstrLog() and addGenConstrExp(). These are available at Gurobi's reference guide. Solvers won't have a problem with 2nd degree $x^2$