Divide using synthetic division $$ ( -9x^{4}-6x^{3}+6x-5 ) \div ( 3x-2 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrr}2&-9&-6&0&6&-5\\& & -18& -48& -96& \color{black}{-180} \\ \hline &\color{blue}{-9}&\color{blue}{-24}&\color{blue}{-48}&\color{blue}{-90}&\color{orangered}{-185} \end{array} $$The solution is:
$$ \frac{ -9x^{4}-6x^{3}+6x-5 }{ x-2 } = \color{blue}{-9x^{3}-24x^{2}-48x-90} \color{red}{~-~} \frac{ \color{red}{ 185 } }{ x-2 } $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x -2 = 0 $ ( $ x = \color{blue}{ 2 } $ ) at the left.
$$ \begin{array}{c|rrrrr}\color{blue}{2}&-9&-6&0&6&-5\\& & & & & \\ \hline &&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrr}2&\color{orangered}{ -9 }&-6&0&6&-5\\& & & & & \\ \hline &\color{orangered}{-9}&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ \left( -9 \right) } = \color{blue}{ -18 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{2}&-9&-6&0&6&-5\\& & \color{blue}{-18} & & & \\ \hline &\color{blue}{-9}&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -6 } + \color{orangered}{ \left( -18 \right) } = \color{orangered}{ -24 } $
$$ \begin{array}{c|rrrrr}2&-9&\color{orangered}{ -6 }&0&6&-5\\& & \color{orangered}{-18} & & & \\ \hline &-9&\color{orangered}{-24}&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ \left( -24 \right) } = \color{blue}{ -48 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{2}&-9&-6&0&6&-5\\& & -18& \color{blue}{-48} & & \\ \hline &-9&\color{blue}{-24}&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ \left( -48 \right) } = \color{orangered}{ -48 } $
$$ \begin{array}{c|rrrrr}2&-9&-6&\color{orangered}{ 0 }&6&-5\\& & -18& \color{orangered}{-48} & & \\ \hline &-9&-24&\color{orangered}{-48}&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ \left( -48 \right) } = \color{blue}{ -96 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{2}&-9&-6&0&6&-5\\& & -18& -48& \color{blue}{-96} & \\ \hline &-9&-24&\color{blue}{-48}&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 6 } + \color{orangered}{ \left( -96 \right) } = \color{orangered}{ -90 } $
$$ \begin{array}{c|rrrrr}2&-9&-6&0&\color{orangered}{ 6 }&-5\\& & -18& -48& \color{orangered}{-96} & \\ \hline &-9&-24&-48&\color{orangered}{-90}& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ \left( -90 \right) } = \color{blue}{ -180 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{2}&-9&-6&0&6&-5\\& & -18& -48& -96& \color{blue}{-180} \\ \hline &-9&-24&-48&\color{blue}{-90}& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ -5 } + \color{orangered}{ \left( -180 \right) } = \color{orangered}{ -185 } $
$$ \begin{array}{c|rrrrr}2&-9&-6&0&6&\color{orangered}{ -5 }\\& & -18& -48& -96& \color{orangered}{-180} \\ \hline &\color{blue}{-9}&\color{blue}{-24}&\color{blue}{-48}&\color{blue}{-90}&\color{orangered}{-185} \end{array} $$Bottom line represents the quotient $ \color{blue}{ -9x^{3}-24x^{2}-48x-90 } $ with a remainder of $ \color{red}{ -185 } $.