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?

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 extremum $(x_1, f(x_1))$, $(x_2, f(x_2))$ 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?