Prove $S$ is composite HINTS ONLY

Let $a, b, c, d, e, f$ be positive integers. Suppose that $S = a + b + c + d + e + f$
  divides both $abc + def $ and $ab + bc + ca − de − ef − fd$. Prove that $S$ is composite.

Must be solved using polynomials in some way
HINTS ONLY
I was thinking of defining $P(x) = (x + a)(x + b)(x+c) + (x + d)(x + e)(x + f)$ such that $P(0) = abc + def$. And $S | P(0)$.
But besides that, I dont see anything obvious.
Hints please?

Hint: $S$ divides all coefficients of $f(x)=(x+a)(x+b)(x+c)-(x-d)(x-e)(x-f)$, so it divides $f(d)$. What would happen if $S$ was prime?
