Divide using synthetic division $$ ( 6x^{4}-13x^{3}-9x^{2}+16x-32 ) \div ( 2x-5 ) $$
The synthetic division table is:
$$ \begin{array}{c|rrrrr}5&6&-13&-9&16&-32\\& & 30& 85& 380& \color{black}{1980} \\ \hline &\color{blue}{6}&\color{blue}{17}&\color{blue}{76}&\color{blue}{396}&\color{orangered}{1948} \end{array} $$The solution is:
$$ \dfrac{ 6x^{4}-13x^{3}-9x^{2}+16x-32 }{ x-5 } = \color{blue}{6x^{3}+17x^{2}+76x+396} ~+~ \dfrac{ \color{red}{ 1948 } }{ x-5 } $$Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x -5 = 0 $ ( $ x = \color{blue}{ 5 } $ ) at the left.
$$ \begin{array}{c|rrrrr}\color{blue}{5}&6&-13&-9&16&-32\\& & & & & \\ \hline &&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrr}5&\color{orangered}{ 6 }&-13&-9&16&-32\\& & & & & \\ \hline &\color{orangered}{6}&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 5 } \cdot \color{blue}{ 6 } = \color{blue}{ 30 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{5}&6&-13&-9&16&-32\\& & \color{blue}{30} & & & \\ \hline &\color{blue}{6}&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -13 } + \color{orangered}{ 30 } = \color{orangered}{ 17 } $
$$ \begin{array}{c|rrrrr}5&6&\color{orangered}{ -13 }&-9&16&-32\\& & \color{orangered}{30} & & & \\ \hline &6&\color{orangered}{17}&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 5 } \cdot \color{blue}{ 17 } = \color{blue}{ 85 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{5}&6&-13&-9&16&-32\\& & 30& \color{blue}{85} & & \\ \hline &6&\color{blue}{17}&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -9 } + \color{orangered}{ 85 } = \color{orangered}{ 76 } $
$$ \begin{array}{c|rrrrr}5&6&-13&\color{orangered}{ -9 }&16&-32\\& & 30& \color{orangered}{85} & & \\ \hline &6&17&\color{orangered}{76}&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 5 } \cdot \color{blue}{ 76 } = \color{blue}{ 380 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{5}&6&-13&-9&16&-32\\& & 30& 85& \color{blue}{380} & \\ \hline &6&17&\color{blue}{76}&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 16 } + \color{orangered}{ 380 } = \color{orangered}{ 396 } $
$$ \begin{array}{c|rrrrr}5&6&-13&-9&\color{orangered}{ 16 }&-32\\& & 30& 85& \color{orangered}{380} & \\ \hline &6&17&76&\color{orangered}{396}& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 5 } \cdot \color{blue}{ 396 } = \color{blue}{ 1980 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{5}&6&-13&-9&16&-32\\& & 30& 85& 380& \color{blue}{1980} \\ \hline &6&17&76&\color{blue}{396}& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ -32 } + \color{orangered}{ 1980 } = \color{orangered}{ 1948 } $
$$ \begin{array}{c|rrrrr}5&6&-13&-9&16&\color{orangered}{ -32 }\\& & 30& 85& 380& \color{orangered}{1980} \\ \hline &\color{blue}{6}&\color{blue}{17}&\color{blue}{76}&\color{blue}{396}&\color{orangered}{1948} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 6x^{3}+17x^{2}+76x+396 } $ with a remainder of $ \color{red}{ 1948 } $.