I know that, the function f(x) = (5x^2 + 8x+ 2)/(2x^2 + 6x + 5) sastifying the conditions:

1. The solutions of the f'(x)=0 are -2 and -1;
2. f(-2) = 6 and f(-1) = -1.

How can I find six integer numbers a, b, c, d, e, m from 1 to 10 so that the function
f(x) = (a*x^2 + b*x + c)/(d*x^2 + e*x + m)
so that the equation f'(x)= 0 has two integer solutions x1, x2 and f(x1); f(x2) are also  two integer numbers?

I find by my hand some equations have four integer solutions.

How can I tell Maple to do this? For what the values of integer numbers k, m, n, a, b, c, d so the equation
k/(x^2 + a x  + b) +  m/(x^2 + a x  + c) + n/(x^2 + a x  + d) = 0 have four integer solutions?

EDIT 
x*f'(x^2)  + g'(x) = cos(x)  - 3x^3 and f(x^2) + g(x) = sin(x) -x^4. 

How to find the function f like this:
x*f'(x)  + g'(x) = cos(x)  - 3x^3 and f(x^2) + g(x) = sin(x) -x^4. 
I know that, f(x) = -1/2*x^2 + C. But, I tried. This answer incorrec. 
restart;
f := x -> -1/2*x^2 + C;
g := x -> sin(x) - 1/2*x^4 - C;
diff(f(x^2), x);
f(x^2) + g(x);
x*diff(f(x^2), x) + diff(g(x), x);

I know that the angle between two vectors u = (2, 1, 1) and v = (9, -1, 4) equal to 30 degree. How to find the some options of two vectors u = (a, b, c) and v = (x, y, z), where a, b, c, x, y, z are six integer numbers so that  the angle between two vectors u and v equal to 30 degree?

Can this equation has four integer solutions? 
I am consider the equation a^2*x^2/(x-a)^2+x^2 = m/n and trying to find the integer numbers a, m, n so that the given equation has integer solutions?

Four_integer_solutions.mw