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