Answer
$= 5x^2+x+3+\dfrac{x+7}{3x^2-1}$
Work Step by Step
$ \begin{array}{llllllllll}
& & 5x^{2} & +x & +3 & & & \color{red}{ \small{quotient}} & & \\
& & -- & -- & -- & -- & -- & -- & & \\
3x^{2}-1 & ) & 15x^{4} & +3x^{3} & +4x^{2} & & +4 & \color{blue}{\leftarrow \small{ no\ a_{1}x^{1}... } } & & \\
& & 15x^{4} & & -5x^{2} & & & \color{red}{\leftarrow \small{ 5x^{2}(3x^{2}-1) } } & & \\
& & -- & -- & -- & & & \color{red}{\leftarrow \small{ subtract } } & & \\
& & & 3x^{3} & +9x^{2} & & +4 & & & \\
& & & 3x^{3} & & -x & & \color{red}{\leftarrow \small{x(3x^{2}-1) } } & & \\
& & & -- & -- & -- & & \color{red}{\leftarrow \small{ subtract }} & & \\
& & & & 9x^{2} & +x & +4 & & & \\
& & & & 9x^{2} & & -3 & \color{red}{\leftarrow \small{ 4(3x^{2}-1) } } & & \\
& & & & -- & -- & -- & \color{red}{\leftarrow \small{ subtract } } & & \\
& & & & & & & & & \\
& & & & & x & +7 & \color{red}{\leftarrow \small{ Remainder } } & &
\end{array} $
$(15x^{4}+3x^{3}+4x^{2}+4)\div(3x^{2}-1)=$
$= 5x^2+x+3+\dfrac{x+7}{3x^2-1}$
