Divide using synthetic division $$ ( x^{5}+3x^{4}-46x^{3}-47x^{2}-8x+12 ) \div ( x-6 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrrr}6&1&3&-46&-47&-8&12\\& & 6& 54& 48& 6& \color{black}{-12} \\ \hline &\color{blue}{1}&\color{blue}{9}&\color{blue}{8}&\color{blue}{1}&\color{blue}{-2}&\color{orangered}{0} \end{array} $$The solution is:
$$ \frac{ x^{5}+3x^{4}-46x^{3}-47x^{2}-8x+12 }{ x-6 } = \color{blue}{x^{4}+9x^{3}+8x^{2}+x-2} $$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.
$$ \begin{array}{c|rrrrrr}\color{blue}{6}&1&3&-46&-47&-8&12\\& & & & & & \\ \hline &&&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrrr}6&\color{orangered}{ 1 }&3&-46&-47&-8&12\\& & & & & & \\ \hline &\color{orangered}{1}&&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 6 } \cdot \color{blue}{ 1 } = \color{blue}{ 6 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{6}&1&3&-46&-47&-8&12\\& & \color{blue}{6} & & & & \\ \hline &\color{blue}{1}&&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 3 } + \color{orangered}{ 6 } = \color{orangered}{ 9 } $
$$ \begin{array}{c|rrrrrr}6&1&\color{orangered}{ 3 }&-46&-47&-8&12\\& & \color{orangered}{6} & & & & \\ \hline &1&\color{orangered}{9}&&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 6 } \cdot \color{blue}{ 9 } = \color{blue}{ 54 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{6}&1&3&-46&-47&-8&12\\& & 6& \color{blue}{54} & & & \\ \hline &1&\color{blue}{9}&&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -46 } + \color{orangered}{ 54 } = \color{orangered}{ 8 } $
$$ \begin{array}{c|rrrrrr}6&1&3&\color{orangered}{ -46 }&-47&-8&12\\& & 6& \color{orangered}{54} & & & \\ \hline &1&9&\color{orangered}{8}&&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 6 } \cdot \color{blue}{ 8 } = \color{blue}{ 48 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{6}&1&3&-46&-47&-8&12\\& & 6& 54& \color{blue}{48} & & \\ \hline &1&9&\color{blue}{8}&&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -47 } + \color{orangered}{ 48 } = \color{orangered}{ 1 } $
$$ \begin{array}{c|rrrrrr}6&1&3&-46&\color{orangered}{ -47 }&-8&12\\& & 6& 54& \color{orangered}{48} & & \\ \hline &1&9&8&\color{orangered}{1}&& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 6 } \cdot \color{blue}{ 1 } = \color{blue}{ 6 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{6}&1&3&-46&-47&-8&12\\& & 6& 54& 48& \color{blue}{6} & \\ \hline &1&9&8&\color{blue}{1}&& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ -8 } + \color{orangered}{ 6 } = \color{orangered}{ -2 } $
$$ \begin{array}{c|rrrrrr}6&1&3&-46&-47&\color{orangered}{ -8 }&12\\& & 6& 54& 48& \color{orangered}{6} & \\ \hline &1&9&8&1&\color{orangered}{-2}& \end{array} $$Step 10 : 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 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{6}&1&3&-46&-47&-8&12\\& & 6& 54& 48& 6& \color{blue}{-12} \\ \hline &1&9&8&1&\color{blue}{-2}& \end{array} $$Step 11 : Add down last column: $ \color{orangered}{ 12 } + \color{orangered}{ \left( -12 \right) } = \color{orangered}{ 0 } $
$$ \begin{array}{c|rrrrrr}6&1&3&-46&-47&-8&\color{orangered}{ 12 }\\& & 6& 54& 48& 6& \color{orangered}{-12} \\ \hline &\color{blue}{1}&\color{blue}{9}&\color{blue}{8}&\color{blue}{1}&\color{blue}{-2}&\color{orangered}{0} \end{array} $$Bottom line represents the quotient $ \color{blue}{ x^{4}+9x^{3}+8x^{2}+x-2 } $ with a remainder of $ \color{red}{ 0 } $.