Divide using synthetic division $$ ( x^{5}+5 ) \div ( x+4 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrrr}-4&1&0&0&0&0&5\\& & -4& 16& -64& 256& \color{black}{-1024} \\ \hline &\color{blue}{1}&\color{blue}{-4}&\color{blue}{16}&\color{blue}{-64}&\color{blue}{256}&\color{orangered}{-1019} \end{array} $$The solution is:
$$ \frac{ x^{5}+5 }{ x+4 } = \color{blue}{x^{4}-4x^{3}+16x^{2}-64x+256} \color{red}{~-~} \frac{ \color{red}{ 1019 } }{ x+4 } $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x + 4 = 0 $ ( $ x = \color{blue}{ -4 } $ ) at the left.
$$ \begin{array}{c|rrrrrr}\color{blue}{-4}&1&0&0&0&0&5\\& & & & & & \\ \hline &&&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrrr}-4&\color{orangered}{ 1 }&0&0&0&0&5\\& & & & & & \\ \hline &\color{orangered}{1}&&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -4 } \cdot \color{blue}{ 1 } = \color{blue}{ -4 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{-4}&1&0&0&0&0&5\\& & \color{blue}{-4} & & & & \\ \hline &\color{blue}{1}&&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ \left( -4 \right) } = \color{orangered}{ -4 } $
$$ \begin{array}{c|rrrrrr}-4&1&\color{orangered}{ 0 }&0&0&0&5\\& & \color{orangered}{-4} & & & & \\ \hline &1&\color{orangered}{-4}&&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -4 } \cdot \color{blue}{ \left( -4 \right) } = \color{blue}{ 16 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{-4}&1&0&0&0&0&5\\& & -4& \color{blue}{16} & & & \\ \hline &1&\color{blue}{-4}&&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 16 } = \color{orangered}{ 16 } $
$$ \begin{array}{c|rrrrrr}-4&1&0&\color{orangered}{ 0 }&0&0&5\\& & -4& \color{orangered}{16} & & & \\ \hline &1&-4&\color{orangered}{16}&&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -4 } \cdot \color{blue}{ 16 } = \color{blue}{ -64 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{-4}&1&0&0&0&0&5\\& & -4& 16& \color{blue}{-64} & & \\ \hline &1&-4&\color{blue}{16}&&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ \left( -64 \right) } = \color{orangered}{ -64 } $
$$ \begin{array}{c|rrrrrr}-4&1&0&0&\color{orangered}{ 0 }&0&5\\& & -4& 16& \color{orangered}{-64} & & \\ \hline &1&-4&16&\color{orangered}{-64}&& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -4 } \cdot \color{blue}{ \left( -64 \right) } = \color{blue}{ 256 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{-4}&1&0&0&0&0&5\\& & -4& 16& -64& \color{blue}{256} & \\ \hline &1&-4&16&\color{blue}{-64}&& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 256 } = \color{orangered}{ 256 } $
$$ \begin{array}{c|rrrrrr}-4&1&0&0&0&\color{orangered}{ 0 }&5\\& & -4& 16& -64& \color{orangered}{256} & \\ \hline &1&-4&16&-64&\color{orangered}{256}& \end{array} $$Step 10 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -4 } \cdot \color{blue}{ 256 } = \color{blue}{ -1024 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{-4}&1&0&0&0&0&5\\& & -4& 16& -64& 256& \color{blue}{-1024} \\ \hline &1&-4&16&-64&\color{blue}{256}& \end{array} $$Step 11 : Add down last column: $ \color{orangered}{ 5 } + \color{orangered}{ \left( -1024 \right) } = \color{orangered}{ -1019 } $
$$ \begin{array}{c|rrrrrr}-4&1&0&0&0&0&\color{orangered}{ 5 }\\& & -4& 16& -64& 256& \color{orangered}{-1024} \\ \hline &\color{blue}{1}&\color{blue}{-4}&\color{blue}{16}&\color{blue}{-64}&\color{blue}{256}&\color{orangered}{-1019} \end{array} $$Bottom line represents the quotient $ \color{blue}{ x^{4}-4x^{3}+16x^{2}-64x+256 } $ with a remainder of $ \color{red}{ -1019 } $.