Answer
$x^{2} +x -2$
Work Step by Step
$\begin{array}{ccccccccccc}
& &x^{2} & +x & -2 & \\
& &--&--&--&--\\
x-3&) & x^{3} &-2x^{2} &-5x& +6 & \\
& & x^{3}&-3x^{2} & & & \color{red}{\leftarrow \small{x^{2}(x-3) } } \\
& &--&-- & & &\color{red}{ \small{subtract}} \\
& & & x^{2} & -5x & +6 & \\
& & & x^{2} & -3x & & \color{red}{\leftarrow \small{x(x-3) } }\\
& & &--&-- & & \color{red}{ \small{subtract}} \\
& & & & -2x & +6 &\\
& & & & -2x & +6 & \color{red}{\leftarrow \small{-2(x-3) } }\\
& & & &-- &-- & \color{red}{ \small{subtract}} \\
& & & & & 0
\end{array}$
Quotient = $x^{2} +x -2$
Remainder = $0$.
$\displaystyle \frac{dividend}{divisor}=quotient+\frac{remainder}{divisor}$
$\displaystyle \frac{x^{3} -2x^{2} -5x +6}{x-3}$ = $x^{2} +x -2$
