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

I  am trying to find the integer numbers a, b, c, d, m so that the equation

(x−a)(x−b)(x−c)(x−d)=m

where a+d=b+c  and m≠0 has four integer solutions.

I found this equation randomly
solve((x-5) (x-3) (x+6) (x+8)=504, x);
How to find number a, b, c, d, m so that the equation
(x−a)(x−b)(x−c)(x−d)=m
has four integer solutions?