Answer
$x^{2}+5x-3+\displaystyle \frac{3}{2x+3}$
Work Step by Step
$\begin{array}{ccccccccccc}
& &x^{2} & +5x & -3 & \\
& &--&-- &--& \\
2x+3&) & 2x^{3} &+13x^{2}& +9x& -6 & & \\
& & 2x^{3} & +3x^{2 }& & & \color{red}{\leftarrow \small{x^{2}(2x+3) } } \\
& &--&-- & & & \color{red}{ \small{subtract}} \\
& & & 10x^{2}& +9x& -6 & & \\
& & & 10x^{2} & +15x& & \color{red}{\leftarrow \small{5x(2x+3) } }\\
& & &--&-- & &\color{red}{ \small{subtract}} \\
& & & & -6x & -6 & \\
& & & & -6x & -9 & \color{red}{\leftarrow \small{-3(2x+3) } }\\
& & & &-- & -- &\color{red}{ \small{subtract}} \\
& & & & & 3 &
\end{array}$
Quotient = $x^{2}+5x-3$
Remainder = $ 3$.
$\displaystyle \frac{dividend}{divisor}=quotient+\frac{remainder}{divisor}$
$\displaystyle \frac{ 2x^{3}+13x^{2}+9x-6}{2x+3}$ = $x^{2}+5x-3+\displaystyle \frac{3}{2x+3}$
