How can I find the integer numbers `a, b, c, m,n` so that the funtion
`f[x]=(a x^2 + b x + c)/(m x + n)` with the equation
`D[f[x], x]] == 0` have two integer solutions `x1`, `x2` and the values `f[x1]` and `f[x2]` are also integer numbers?

I tried 

    Clear["Global`*"]
    f[x_] = (a x^2 + b x + c)/(m x + n);
    x1 = (-a n - Sqrt[a c m^2 - a b m n + a^2 n^2])/(a m);
    x2 = (-a n + Sqrt[a c m^2 - a b m n + a^2 n^2])/(a m);
    list = Table[
      If[GCD[a, b, c] == 1 && GCD[m, n] == 1 && 
        a c m^2 - a b m n + a^2 n^2 > 0 && a n - b m  != 0 && 
        a (c m^2 - b m n + a n^2) > 0 && b^2 - 4 * a * c < 0 && 
        IntegerQ[x1] && IntegerQ[x2] && b c m n x1 x2 f[x1]* f[x2] != 0 &&
         IntegerQ[f[x1]] && 
        IntegerQ[f[x2]], {(a x^2 + b x  + c)/(m x + n), {x1, f[x1]}, {x2, 
         f[x2]}}, Nothing], {a, 1, 10}, {b, 1, 10}, {c, 1, 10}, {m, 1, 10}, {n, 1, 10}]

[![enter image description here][1]][1]

How can I remove the empty set in my results?

  [1]: https://i.sstatic.net/ufn5S.png