The function Resultant
is good for this type of problem (though this can be sloooow..). After changing variables to lower case:
Timing[res = Resultant[polys[[1]], polys[[2]], g];]
(* Out[30]= {1918.83, Null} *)
It's a big expression.
In[31]:= LeafCount[res]
(* Out[31]= 321649 *)
It can be factored (`FactorSquareFree is what I would usually do first though, as it is faster and, in this case, gives the same result).
Timing[res2 = FactorList[res];]
(* Out[32]= {29.9531, Null} *)
The first two are probably not useful factors (I checked leaf counts before deciding this).
Most[res2]
(* Out[48]= {{95367431640625, 1}, {a^5 - f - 5 a r, 15}} *)
The remaining factor has squarefree and has leaf count of "only" around 70K. It's the one of interest, in terms of the elimination problem at hand.