Divide using synthetic division $$ ( x^{4}-3x^{3}-47x^{2}+37x-21 ) \div ( x+6 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrr}-6&1&-3&-47&37&-21\\& & -6& 54& -42& \color{black}{30} \\ \hline &\color{blue}{1}&\color{blue}{-9}&\color{blue}{7}&\color{blue}{-5}&\color{orangered}{9} \end{array} $$The solution is:
$$ \frac{ x^{4}-3x^{3}-47x^{2}+37x-21 }{ x+6 } = \color{blue}{x^{3}-9x^{2}+7x-5} ~+~ \frac{ \color{red}{ 9 } }{ x+6 } $$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.
$$ \begin{array}{c|rrrrr}\color{blue}{-6}&1&-3&-47&37&-21\\& & & & & \\ \hline &&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrr}-6&\color{orangered}{ 1 }&-3&-47&37&-21\\& & & & & \\ \hline &\color{orangered}{1}&&&& \end{array} $$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 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-6}&1&-3&-47&37&-21\\& & \color{blue}{-6} & & & \\ \hline &\color{blue}{1}&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -3 } + \color{orangered}{ \left( -6 \right) } = \color{orangered}{ -9 } $
$$ \begin{array}{c|rrrrr}-6&1&\color{orangered}{ -3 }&-47&37&-21\\& & \color{orangered}{-6} & & & \\ \hline &1&\color{orangered}{-9}&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -6 } \cdot \color{blue}{ \left( -9 \right) } = \color{blue}{ 54 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-6}&1&-3&-47&37&-21\\& & -6& \color{blue}{54} & & \\ \hline &1&\color{blue}{-9}&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -47 } + \color{orangered}{ 54 } = \color{orangered}{ 7 } $
$$ \begin{array}{c|rrrrr}-6&1&-3&\color{orangered}{ -47 }&37&-21\\& & -6& \color{orangered}{54} & & \\ \hline &1&-9&\color{orangered}{7}&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -6 } \cdot \color{blue}{ 7 } = \color{blue}{ -42 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-6}&1&-3&-47&37&-21\\& & -6& 54& \color{blue}{-42} & \\ \hline &1&-9&\color{blue}{7}&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 37 } + \color{orangered}{ \left( -42 \right) } = \color{orangered}{ -5 } $
$$ \begin{array}{c|rrrrr}-6&1&-3&-47&\color{orangered}{ 37 }&-21\\& & -6& 54& \color{orangered}{-42} & \\ \hline &1&-9&7&\color{orangered}{-5}& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -6 } \cdot \color{blue}{ \left( -5 \right) } = \color{blue}{ 30 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-6}&1&-3&-47&37&-21\\& & -6& 54& -42& \color{blue}{30} \\ \hline &1&-9&7&\color{blue}{-5}& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ -21 } + \color{orangered}{ 30 } = \color{orangered}{ 9 } $
$$ \begin{array}{c|rrrrr}-6&1&-3&-47&37&\color{orangered}{ -21 }\\& & -6& 54& -42& \color{orangered}{30} \\ \hline &\color{blue}{1}&\color{blue}{-9}&\color{blue}{7}&\color{blue}{-5}&\color{orangered}{9} \end{array} $$Bottom line represents the quotient $ \color{blue}{ x^{3}-9x^{2}+7x-5 } $ with a remainder of $ \color{red}{ 9 } $.