Divide using synthetic division $$ ( x^{6}-16x^{5}+31x^{3}+21x^{2}+6x-1 ) \div ( x+1 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrrrr}-1&1&-16&0&31&21&6&-1\\& & -1& 17& -17& -14& -7& \color{black}{1} \\ \hline &\color{blue}{1}&\color{blue}{-17}&\color{blue}{17}&\color{blue}{14}&\color{blue}{7}&\color{blue}{-1}&\color{orangered}{0} \end{array} $$The solution is:
$$ \frac{ x^{6}-16x^{5}+31x^{3}+21x^{2}+6x-1 }{ x+1 } = \color{blue}{x^{5}-17x^{4}+17x^{3}+14x^{2}+7x-1} $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x + 1 = 0 $ ( $ x = \color{blue}{ -1 } $ ) at the left.
$$ \begin{array}{c|rrrrrrr}\color{blue}{-1}&1&-16&0&31&21&6&-1\\& & & & & & & \\ \hline &&&&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrrrr}-1&\color{orangered}{ 1 }&-16&0&31&21&6&-1\\& & & & & & & \\ \hline &\color{orangered}{1}&&&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -1 } \cdot \color{blue}{ 1 } = \color{blue}{ -1 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{-1}&1&-16&0&31&21&6&-1\\& & \color{blue}{-1} & & & & & \\ \hline &\color{blue}{1}&&&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -16 } + \color{orangered}{ \left( -1 \right) } = \color{orangered}{ -17 } $
$$ \begin{array}{c|rrrrrrr}-1&1&\color{orangered}{ -16 }&0&31&21&6&-1\\& & \color{orangered}{-1} & & & & & \\ \hline &1&\color{orangered}{-17}&&&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -1 } \cdot \color{blue}{ \left( -17 \right) } = \color{blue}{ 17 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{-1}&1&-16&0&31&21&6&-1\\& & -1& \color{blue}{17} & & & & \\ \hline &1&\color{blue}{-17}&&&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 17 } = \color{orangered}{ 17 } $
$$ \begin{array}{c|rrrrrrr}-1&1&-16&\color{orangered}{ 0 }&31&21&6&-1\\& & -1& \color{orangered}{17} & & & & \\ \hline &1&-17&\color{orangered}{17}&&&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -1 } \cdot \color{blue}{ 17 } = \color{blue}{ -17 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{-1}&1&-16&0&31&21&6&-1\\& & -1& 17& \color{blue}{-17} & & & \\ \hline &1&-17&\color{blue}{17}&&&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 31 } + \color{orangered}{ \left( -17 \right) } = \color{orangered}{ 14 } $
$$ \begin{array}{c|rrrrrrr}-1&1&-16&0&\color{orangered}{ 31 }&21&6&-1\\& & -1& 17& \color{orangered}{-17} & & & \\ \hline &1&-17&17&\color{orangered}{14}&&& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -1 } \cdot \color{blue}{ 14 } = \color{blue}{ -14 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{-1}&1&-16&0&31&21&6&-1\\& & -1& 17& -17& \color{blue}{-14} & & \\ \hline &1&-17&17&\color{blue}{14}&&& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ 21 } + \color{orangered}{ \left( -14 \right) } = \color{orangered}{ 7 } $
$$ \begin{array}{c|rrrrrrr}-1&1&-16&0&31&\color{orangered}{ 21 }&6&-1\\& & -1& 17& -17& \color{orangered}{-14} & & \\ \hline &1&-17&17&14&\color{orangered}{7}&& \end{array} $$Step 10 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -1 } \cdot \color{blue}{ 7 } = \color{blue}{ -7 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{-1}&1&-16&0&31&21&6&-1\\& & -1& 17& -17& -14& \color{blue}{-7} & \\ \hline &1&-17&17&14&\color{blue}{7}&& \end{array} $$Step 11 : Add down last column: $ \color{orangered}{ 6 } + \color{orangered}{ \left( -7 \right) } = \color{orangered}{ -1 } $
$$ \begin{array}{c|rrrrrrr}-1&1&-16&0&31&21&\color{orangered}{ 6 }&-1\\& & -1& 17& -17& -14& \color{orangered}{-7} & \\ \hline &1&-17&17&14&7&\color{orangered}{-1}& \end{array} $$Step 12 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -1 } \cdot \color{blue}{ \left( -1 \right) } = \color{blue}{ 1 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{-1}&1&-16&0&31&21&6&-1\\& & -1& 17& -17& -14& -7& \color{blue}{1} \\ \hline &1&-17&17&14&7&\color{blue}{-1}& \end{array} $$Step 13 : Add down last column: $ \color{orangered}{ -1 } + \color{orangered}{ 1 } = \color{orangered}{ 0 } $
$$ \begin{array}{c|rrrrrrr}-1&1&-16&0&31&21&6&\color{orangered}{ -1 }\\& & -1& 17& -17& -14& -7& \color{orangered}{1} \\ \hline &\color{blue}{1}&\color{blue}{-17}&\color{blue}{17}&\color{blue}{14}&\color{blue}{7}&\color{blue}{-1}&\color{orangered}{0} \end{array} $$Bottom line represents the quotient $ \color{blue}{ x^{5}-17x^{4}+17x^{3}+14x^{2}+7x-1 } $ with a remainder of $ \color{red}{ 0 } $.