I know that, the function f[x_] := (x^2 + 6 x + 5)/(x^2 + 2 x + 3) with D[f[x], x] have two integer solutions -2, 1 and f[-2]=1, f[1]=2. My code

f[x_] := (x^2 + 6 x + 5)/(x^2 + 2 x + 3)
Reduce[Simplify[D[f[x], x]] == 0, x]
f[-2]
f[1]

How can I find the integer numbers a, b, c, d, e, m so that the funtion f[x]=(a x^2 + b x + c)/(d x^2 + e x + m) 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?