Hello, Ununuquantium!
Didn't you Preview your work before you posted it?
If you add these equations you get:
(a-2)(a-1)a(a+1)(a+2) + (b-2)(b-1)b(b+1)(b-2) + (c-2)(c-1)c(c+1)(d+1) = 0
One solution is: a=b=c=2 or 1 or 0 or -1 or -2
I don't know if there are other solutions, maybe someone can proof that a,b,c must be equal?