Answer
$x^{4}-x^{2}+x+1+\displaystyle \frac{3}{x-2}$
Work Step by Step
Dividing with $x-c\qquad...\qquad c=2.$
$\begin{array}{rrrrrrrr}
\underline{2}| &1 & -2& -1 & 3 & -1 & 1 & & \\
& & 2 & 0 & -2 & 2 & 2 & & \\
& --& -- &-- &-- &-- & -- & & \\
&1 & 0 & -1 & 1 & 1 & 3 & & \end{array}$
Quotient = $ x^{4}-x^{2}+x+1$
Remainder = $3$
$\displaystyle \frac{dividend}{divisor}=quotient+\frac{remainder}{divisor}$
$\displaystyle \frac{x^{5}-2x^{4}-x^{3}+3x^{2}-x+1}{x-2}$ = $x^{4}-x^{2}+x+1+\displaystyle \frac{3}{x-2}$
