Divide using synthetic division $$ ( x^{6}-5x^{5}-x^{4}+21x^{3}+4x^{2}-28x-16 ) \div ( x+1 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrrrr}-1&1&-5&-1&21&4&-28&-16\\& & -1& 6& -5& -16& 12& \color{black}{16} \\ \hline &\color{blue}{1}&\color{blue}{-6}&\color{blue}{5}&\color{blue}{16}&\color{blue}{-12}&\color{blue}{-16}&\color{orangered}{0} \end{array} $$The solution is:
$$ \frac{ x^{6}-5x^{5}-x^{4}+21x^{3}+4x^{2}-28x-16 }{ x+1 } = \color{blue}{x^{5}-6x^{4}+5x^{3}+16x^{2}-12x-16} $$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.
$$ \begin{array}{c|rrrrrrr}\color{blue}{-1}&1&-5&-1&21&4&-28&-16\\& & & & & & & \\ \hline &&&&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrrrr}-1&\color{orangered}{ 1 }&-5&-1&21&4&-28&-16\\& & & & & & & \\ \hline &\color{orangered}{1}&&&&&& \end{array} $$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 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{-1}&1&-5&-1&21&4&-28&-16\\& & \color{blue}{-1} & & & & & \\ \hline &\color{blue}{1}&&&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -5 } + \color{orangered}{ \left( -1 \right) } = \color{orangered}{ -6 } $
$$ \begin{array}{c|rrrrrrr}-1&1&\color{orangered}{ -5 }&-1&21&4&-28&-16\\& & \color{orangered}{-1} & & & & & \\ \hline &1&\color{orangered}{-6}&&&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -1 } \cdot \color{blue}{ \left( -6 \right) } = \color{blue}{ 6 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{-1}&1&-5&-1&21&4&-28&-16\\& & -1& \color{blue}{6} & & & & \\ \hline &1&\color{blue}{-6}&&&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -1 } + \color{orangered}{ 6 } = \color{orangered}{ 5 } $
$$ \begin{array}{c|rrrrrrr}-1&1&-5&\color{orangered}{ -1 }&21&4&-28&-16\\& & -1& \color{orangered}{6} & & & & \\ \hline &1&-6&\color{orangered}{5}&&&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -1 } \cdot \color{blue}{ 5 } = \color{blue}{ -5 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{-1}&1&-5&-1&21&4&-28&-16\\& & -1& 6& \color{blue}{-5} & & & \\ \hline &1&-6&\color{blue}{5}&&&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 21 } + \color{orangered}{ \left( -5 \right) } = \color{orangered}{ 16 } $
$$ \begin{array}{c|rrrrrrr}-1&1&-5&-1&\color{orangered}{ 21 }&4&-28&-16\\& & -1& 6& \color{orangered}{-5} & & & \\ \hline &1&-6&5&\color{orangered}{16}&&& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -1 } \cdot \color{blue}{ 16 } = \color{blue}{ -16 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{-1}&1&-5&-1&21&4&-28&-16\\& & -1& 6& -5& \color{blue}{-16} & & \\ \hline &1&-6&5&\color{blue}{16}&&& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ 4 } + \color{orangered}{ \left( -16 \right) } = \color{orangered}{ -12 } $
$$ \begin{array}{c|rrrrrrr}-1&1&-5&-1&21&\color{orangered}{ 4 }&-28&-16\\& & -1& 6& -5& \color{orangered}{-16} & & \\ \hline &1&-6&5&16&\color{orangered}{-12}&& \end{array} $$Step 10 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -1 } \cdot \color{blue}{ \left( -12 \right) } = \color{blue}{ 12 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{-1}&1&-5&-1&21&4&-28&-16\\& & -1& 6& -5& -16& \color{blue}{12} & \\ \hline &1&-6&5&16&\color{blue}{-12}&& \end{array} $$Step 11 : Add down last column: $ \color{orangered}{ -28 } + \color{orangered}{ 12 } = \color{orangered}{ -16 } $
$$ \begin{array}{c|rrrrrrr}-1&1&-5&-1&21&4&\color{orangered}{ -28 }&-16\\& & -1& 6& -5& -16& \color{orangered}{12} & \\ \hline &1&-6&5&16&-12&\color{orangered}{-16}& \end{array} $$Step 12 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -1 } \cdot \color{blue}{ \left( -16 \right) } = \color{blue}{ 16 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{-1}&1&-5&-1&21&4&-28&-16\\& & -1& 6& -5& -16& 12& \color{blue}{16} \\ \hline &1&-6&5&16&-12&\color{blue}{-16}& \end{array} $$Step 13 : Add down last column: $ \color{orangered}{ -16 } + \color{orangered}{ 16 } = \color{orangered}{ 0 } $
$$ \begin{array}{c|rrrrrrr}-1&1&-5&-1&21&4&-28&\color{orangered}{ -16 }\\& & -1& 6& -5& -16& 12& \color{orangered}{16} \\ \hline &\color{blue}{1}&\color{blue}{-6}&\color{blue}{5}&\color{blue}{16}&\color{blue}{-12}&\color{blue}{-16}&\color{orangered}{0} \end{array} $$Bottom line represents the quotient $ \color{blue}{ x^{5}-6x^{4}+5x^{3}+16x^{2}-12x-16 } $ with a remainder of $ \color{red}{ 0 } $.