Graph of the fuction $y=\dfrac{(x-26)(x+9)}{(x+14)(x+19)}$ has maximum point and minimum point are `(-16,-49)` and  `(-4,-1)` whose coordinates are integer numbers. I am trying to find four integer numbers `a, b, c, d` so that graph of the function $f(x) = \dfrac{(x+a)(x+b)}{(x+c)(x+d)}$ has two extrmum $(x_1, f(x_1))$, $(x_2, f(x_2))$ where $x_1$, $x_2$, $f(x_1)$, $f(x_2)$ are four integer numbers. I tried

    Clear["Global`*"];
    f[x_] = ((x + a) (x + b))/((x + c) (x + d));
    sol = Solve[f'[x] == 0, x] // Simplify;
    x1 = x /. sol[[1]]
    x2 = x /. sol[[2]]
    f[x1] // FullSimplify
    f[x2] // FullSimplify
    Table[{IntegerQ[x1] && IntegerQ[x2] &&  IntegerQ[f[x1]] && 
       IntegerQ[f[x2]]}, {-30 <= a <= 30 && -30 <= b <= 30 && -30 <= c <= 
        30 && -30 <= d <= 30 && a > b  && c > d}]

I can not get the result. How can I get the results?