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?

I see this question at here
https://mathematica.stackexchange.com/questions/239874/how-to-reduce-timing-to-find-the-integer-numbers-a-b-c-d-e-f-g-h-k-m-of

How to solve this problem by Maple?

I am trying to solve this equation, I tried
 

restart; 
ListTools[Categorize]; 
L := []; 
for a to 20 do
 for b to 20 do 
for c to 20 do 
for d to 20 do 
for e to 20 do 
for f to 20 do 
for g to 20 do 
for h to 20 do 
if sqrt(a+b*sqrt(c+d*sqrt(e +f*sqrt(g)))) = h then L := [op(L), [a, b, c, d, e, f, g, h]] end if
 end do end do end do end do end do end do end do end do; nops(L)

where g is not a square of a integer number . I do not get any solutions for a long time. How to reduce timing to solve this equation?

 

I know that, tetrahedron with vertices (0, 0, 0), (3, 0, 0), (0, 5, 0), (0, 0, 14) has center of insphere is  (1, 1, 1) and radius R = 1. How to create a tetrahedron whose coordinates of all vertices are integers, the inscribed spherical center and the inscribed spherical radius are integers?