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