jmc

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