Answer
$(\displaystyle \frac{f}{g})(x)= x^3-x^2+x-2$
Work Step by Step
$(\displaystyle \frac{f}{g})(x)= \frac{2x^{4}-7x^{3}+7x^{2}-9x+10}{2x-5}$
$ \begin{array}{llllllllll}
& & x^{3} & -x^{2} & +x & -2 & & \color{red}{ \small{Quotient}} & & \\
& & -- & -- & -- & -- & -- & -- & & \\
2x-5 & ) & 2x^{4} & -7x^{3} & +7x^{2} & -9x & +10 & & & \\
& & 2x^{4} & -5x^{3} & & & & \color{red}{\leftarrow \small{ x^{3}(2x-5) } } & & \\
& & -- & -- & & & & \color{red}{\leftarrow \small{ subtract } } & & \\
& & & -2x^{2} & +7x^{2} & -9x & +10 & & & \\
& & & -2x^{2} & +5x & & & \color{red}{\leftarrow \small{-x^2(2x-5) } } & & \\
& & & -- & -- & & & \color{red}{\leftarrow \small{ subtract }} & & \\
& & & & 2x & -9x & +10 & & & \\
& & & & 2x & -5x & & \color{red}{\leftarrow \small{+x(2x-5) } } & & \\
& & & & -- & -- & & \color{red}{\leftarrow \small{ subtract } } & & \\
& & & & & -4x & +10 & & & \\
& & & & & -4x & +10 & \color{red}{\leftarrow \small{ -2(2x-5) } } & & \\
& & & & & -- & -- & \color{red}{\leftarrow \small{ subtract } } & & \\
& & & & & & 0 & \color{red}{\leftarrow \small{ Remainder } } & &
\end{array} $
$(\displaystyle \frac{f}{g})(x)= x^3-x^2+x-2$
