Divide using synthetic division $$ ( -2x^{4}+12x^{3}-3x+9 ) \div ( x-6 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrr}6&-2&12&0&-3&9\\& & -12& 0& 0& \color{black}{-18} \\ \hline &\color{blue}{-2}&\color{blue}{0}&\color{blue}{0}&\color{blue}{-3}&\color{orangered}{-9} \end{array} $$The solution is:
$$ \frac{ -2x^{4}+12x^{3}-3x+9 }{ x-6 } = \color{blue}{-2x^{3}-3} \color{red}{~-~} \frac{ \color{red}{ 9 } }{ x-6 } $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x -6 = 0 $ ( $ x = \color{blue}{ 6 } $ ) at the left.
$$ \begin{array}{c|rrrrr}\color{blue}{6}&-2&12&0&-3&9\\& & & & & \\ \hline &&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrr}6&\color{orangered}{ -2 }&12&0&-3&9\\& & & & & \\ \hline &\color{orangered}{-2}&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 6 } \cdot \color{blue}{ \left( -2 \right) } = \color{blue}{ -12 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{6}&-2&12&0&-3&9\\& & \color{blue}{-12} & & & \\ \hline &\color{blue}{-2}&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 12 } + \color{orangered}{ \left( -12 \right) } = \color{orangered}{ 0 } $
$$ \begin{array}{c|rrrrr}6&-2&\color{orangered}{ 12 }&0&-3&9\\& & \color{orangered}{-12} & & & \\ \hline &-2&\color{orangered}{0}&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 6 } \cdot \color{blue}{ 0 } = \color{blue}{ 0 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{6}&-2&12&0&-3&9\\& & -12& \color{blue}{0} & & \\ \hline &-2&\color{blue}{0}&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 0 } = \color{orangered}{ 0 } $
$$ \begin{array}{c|rrrrr}6&-2&12&\color{orangered}{ 0 }&-3&9\\& & -12& \color{orangered}{0} & & \\ \hline &-2&0&\color{orangered}{0}&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 6 } \cdot \color{blue}{ 0 } = \color{blue}{ 0 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{6}&-2&12&0&-3&9\\& & -12& 0& \color{blue}{0} & \\ \hline &-2&0&\color{blue}{0}&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -3 } + \color{orangered}{ 0 } = \color{orangered}{ -3 } $
$$ \begin{array}{c|rrrrr}6&-2&12&0&\color{orangered}{ -3 }&9\\& & -12& 0& \color{orangered}{0} & \\ \hline &-2&0&0&\color{orangered}{-3}& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 6 } \cdot \color{blue}{ \left( -3 \right) } = \color{blue}{ -18 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{6}&-2&12&0&-3&9\\& & -12& 0& 0& \color{blue}{-18} \\ \hline &-2&0&0&\color{blue}{-3}& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ 9 } + \color{orangered}{ \left( -18 \right) } = \color{orangered}{ -9 } $
$$ \begin{array}{c|rrrrr}6&-2&12&0&-3&\color{orangered}{ 9 }\\& & -12& 0& 0& \color{orangered}{-18} \\ \hline &\color{blue}{-2}&\color{blue}{0}&\color{blue}{0}&\color{blue}{-3}&\color{orangered}{-9} \end{array} $$Bottom line represents the quotient $ \color{blue}{ -2x^{3}-3 } $ with a remainder of $ \color{red}{ -9 } $.