Find pair of polynomials a(x) and b(x) If $a(x) + b(x) = x^6-1$ and $\gcd(a(x),b(x))=x+1$ then find a pair of polynomials of $a(x)$,$b(x)$.
Prove or disprove, if there exists more than 1  more distinct values of the polynomials.

Hint $\ \ $ If $\rm\:a\!+\!b = c\:(x\!+\!1)\:$ then 
$$\rm\:x\!+\!1 = (a,b) = (a,c\,(x\!+\!1)\!-\!a) = (a,c\,(x\!+\!1))\iff a = d(x\!+\!1),\,\ (d,c) = 1$$
Thus for $\rm\:c(x\!+\!1) = x^6\!-\!1\:$ it's  true $\rm\iff a = d(x\!+\!1),\:$ for $\rm\,d\,$ coprime to $\rm\:c = (x^6\!-\!1)/(x+1),\:$ or, equivalently, $\rm\,d\,$ coprime to $\rm\:x\!-\!1,\ x^2\pm x + 1$
