Answer
$x^3+x^2-x-3+\dfrac{-x+5}{x^2-x+2}$
Work Step by Step
$ \begin{array}{llllllllll}
& & x^{3} & +x^{2} & -x & -3 & & & \color{red}{ \small{Quotient}} & \\
& & -- & -- & -- & -- & -- & -- & & \\
x^{2}-x+2 & ) & x^{5} & +0\cdot x^{4} & +0x^{3} & +0x^{2} & +0x & -1 & & \\
& & x^{5} & -x^{4} & +2x^{3} & & & & \color{red}{\leftarrow \small{ x^{3}(x^{2}-x+2) } } & \\
& & -- & -- & -- & & & & \color{red}{\leftarrow \small{ subtract } } & \\
& & & x^{4} & -2x^{3} & +0x^{2} & +0x & -1 & & \\
& & & x^{4} & -x^{3} & +2x^{2} & & & \color{red}{\leftarrow \small{x^{2}(x^{2}-x+2) } } & \\
& & & -- & -- & -- & & & \color{red}{\leftarrow \small{ subtract }} & \\
& & & & -x^{3} & -2x^{2} & +0x & -1 & & \\
& & & & -x^{3} & +x^{2} & -2x & & \color{red}{\leftarrow \small{-x(x^{2}-x+2) } } & \\
& & & & -- & -- & -- & & \color{red}{\leftarrow \small{ subtract }} & \\
& & & & & -3x^{2} & +2x & -1 & \color{red}{\leftarrow \small{-3(x^{2}-x+2) } } & \\
& & & & & -3x^{2} & +3x & -6 & & \\
& & & & & -- & -- & -- & \color{red}{\leftarrow \small{ subtract }} & \\
& & & & & & -x & +5 & \color{red}{\leftarrow \small{ Remainder } } &
\end{array}$
$\displaystyle \frac{x^{5}-1}{x^{2}-x+2} $= $x^3+x^2-x-3+\dfrac{-x+5}{x^2-x+2}$
