[tex]x^3=x\cdot x^2[/tex], and [tex]x(x^2-x+1)=x^3-x^2+x[/tex]. Subtracting this from the dividend gives a remainder of
[tex](x^3+1)-(x^3-x^2+x)=x^2-x+1[/tex]
[tex]x^2=1\cdot x^2[/tex], and [tex]1(x^2-x+1)=x^2-x+1[/tex]. Subtracting this from the previous remainder gives a new one of
[tex](x^2+x-1)-(x^2-x+1)=2x-2[/tex]
[tex]x^2[/tex] does not divide [tex]2x[/tex], so we're done, and we've found that
[tex]\dfrac{x^3+1}{x^2-x+1}=x+1+\dfrac{2x-2}{x^2-x+1}[/tex]
i.e. you get a quotient of [tex]x+1[/tex] and a remainder of [tex]2x-2[/tex].
