Answer
$\frac{x^2+x+1}{x+1}$ and $x\ne1.$
Work Step by Step
The product of a binomial sum and a binomial difference is: $(A+B)(A-B)=A^2-B^2$.
The formula for factoring the sum of two cubes is: $A^3+B^3=(A+B)(A^2-AB+B^2)$.
The formula for factoring the difference of two cubes is: $A^3-B^3=(A-B)(A^2+AB+B^2)$.
Hence here: $\frac{x^3-1}{x^2-1}=\frac{x^3-1^3}{x^2-1^2}=\frac{(x-1)(x^2+x+1)}{(x+1)(x-1)}=\frac{x^2+x+1}{x+1}$ and $x\ne1.$
