Answer
$2x-3-\displaystyle \frac{15}{x-2}$
Work Step by Step
$\begin{array}{ccccccccccc}
& &2x & -3 & & \\
& &--&--&--& \\
x-2&) & 2x^{2} & +x& -9 & \\
& & 2x^{2} & +4x& & \color{red}{\leftarrow \small{2x(x-2) } } \\
& &--&-- & & \color{red}{ \small{subtract}} \\
& & & -3x& -9 & \\
& & & -3x & +6& \color{red}{\leftarrow \small{-3(x-2) } }\\
& & &--&-- & \color{red}{ \small{subtract}} \\
& & & & -15 &\\
\end{array}$
Quotient = $2x-3$
Remainder = $-15$.
$\displaystyle \frac{dividend}{divisor}=quotient+\frac{remainder}{divisor}$
$\displaystyle \frac{ 2x^{2} +x -9}{x-2}$ = $2x-3-\displaystyle \frac{15}{x-2}$
