Answer
$3x^3+5x+4-\dfrac{2x}{x^2-2}$
Work Step by Step
$ \begin{array}{llllllllll}
& & 3x^{3} & & +5x & +4 & & & \color{red}{ \small{Quotient}} & \\
& & -- & -- & -- & -- & -- & -- & & \\
x^{2}-2 & ) & 3x^{5} & & -x^{3} & +4x^{2} & -12x & -8 & & \\
& & 3x^{5} & & -6x^{3} & & & & \color{red}{\leftarrow \small{ 3x^{3}(x^{2}-2) } } & \\
& & -- & -- & -- & & & & \color{red}{\leftarrow \small{ subtract } } & \\
& & & & 5x^{3} & +4x^{2} & -12x & -8 & & \\
& & & & 5x^{3} & & -10x & & \color{red}{\leftarrow \small{5x(x^{2}-2) } } & \\
& & & & -- & -- & -- & & \color{red}{\leftarrow \small{ subtract } } & \\
& & & & & 4x^{2} & -2x & -8 & & \\
& & & & & 4x^{2} & & -8 & \color{red}{\leftarrow \small{ 4(x^{2}-2) } } & \\
& & & & & -- & -- & -- & \color{red}{\leftarrow \small{ subtract } } & \\\\
& & & & & & -2x & & \color{red}{\leftarrow \small{ Remainder } } &
\end{array} $
$(3x^{5}-x^{3}+4x^{2}-12x-8)\div(x^{2}-2)=$
$=3x^3+5x+4-\dfrac{2x}{x^2-2}$
