Answer
$6x^2+3x-1+\dfrac{-3x+1}{3x^2+1}$
Work Step by Step
$ \begin{array}{llllllllll}
& & 6x^{2} & +3x & -1 & & & \color{red}{ \small{Quotient}} & & \\
& & -- & -- & -- & -- & -- & -- & & \\
3x^{2}+1 & ) & 18x^{4} & +9x^{3} & +3x^{2} & +0 & +0 & \color{blue}{\leftarrow \small{ no\ a_{1}x^{1}+a_{0}... } } & & \\
& & 18x^{4} & & +6x^{2} & & & \color{red}{\leftarrow \small{ 6x^{2}(3x^{2}+1) } } & & \\
& & -- & -- & -- & & & \color{red}{\leftarrow \small{ subtract } } & & \\
& & & 9x^{3} & -3x^{2} & +0 & +0 & & & \\
& & & 3x^{3} & & +3x & & \color{red}{\leftarrow \small{3x(3x^{2}+1) } } & & \\
& & & -- & -- & -- & & \color{red}{\leftarrow \small{ subtract }} & & \\
& & & & -3x^{2} & -3x & & & & \\
& & & & -3x^{2} & & -1 & \color{red}{\leftarrow \small{ -1(3x^{2}+1) } } & & \\
& & & & -- & -- & -- & \color{red}{\leftarrow \small{ subtract } } & & \\
& & & & & & & & & \\
& & & & & -3x & +1 & \color{red}{\leftarrow \small{ Remainder } } & &
\end{array} $
$(15x^{4}+3x^{3}+4x^{2}+4)\div(3x^{2}-1)=$
$=6x^2+3x-1+\dfrac{-3x+1}{3x^2+1}$
