jmc

We can perform long division. We can also write $$\begin{array}{rl}\frac{x^9+1}{x-1}& =\frac{(x^9-1)+2}{x-1}\\ & =\frac{x^9-1}{x-1}+\frac{2}{x-1}\\ & =x^8+x^7+x^6+x^5+x^4+x^3+x^2+x+1+\frac{2}{x-1}.\end{array}$$Thus, the quotient is $\boxed{x^8+x^7+x^6+x^5+x^4+x^3+x^2+x+1}.$