Divide using synthetic division $$ ( x^{5}-4x^{4}+8x+13 ) \div ( x+4 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrrr}-4&1&-4&0&0&8&13\\& & -4& 32& -128& 512& \color{black}{-2080} \\ \hline &\color{blue}{1}&\color{blue}{-8}&\color{blue}{32}&\color{blue}{-128}&\color{blue}{520}&\color{orangered}{-2067} \end{array} $$The solution is:
$$ \frac{ x^{5}-4x^{4}+8x+13 }{ x+4 } = \color{blue}{x^{4}-8x^{3}+32x^{2}-128x+520} \color{red}{~-~} \frac{ \color{red}{ 2067 } }{ x+4 } $$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.
$$ \begin{array}{c|rrrrrr}\color{blue}{-4}&1&-4&0&0&8&13\\& & & & & & \\ \hline &&&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrrr}-4&\color{orangered}{ 1 }&-4&0&0&8&13\\& & & & & & \\ \hline &\color{orangered}{1}&&&&& \end{array} $$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 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{-4}&1&-4&0&0&8&13\\& & \color{blue}{-4} & & & & \\ \hline &\color{blue}{1}&&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -4 } + \color{orangered}{ \left( -4 \right) } = \color{orangered}{ -8 } $
$$ \begin{array}{c|rrrrrr}-4&1&\color{orangered}{ -4 }&0&0&8&13\\& & \color{orangered}{-4} & & & & \\ \hline &1&\color{orangered}{-8}&&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -4 } \cdot \color{blue}{ \left( -8 \right) } = \color{blue}{ 32 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{-4}&1&-4&0&0&8&13\\& & -4& \color{blue}{32} & & & \\ \hline &1&\color{blue}{-8}&&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 32 } = \color{orangered}{ 32 } $
$$ \begin{array}{c|rrrrrr}-4&1&-4&\color{orangered}{ 0 }&0&8&13\\& & -4& \color{orangered}{32} & & & \\ \hline &1&-8&\color{orangered}{32}&&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -4 } \cdot \color{blue}{ 32 } = \color{blue}{ -128 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{-4}&1&-4&0&0&8&13\\& & -4& 32& \color{blue}{-128} & & \\ \hline &1&-8&\color{blue}{32}&&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ \left( -128 \right) } = \color{orangered}{ -128 } $
$$ \begin{array}{c|rrrrrr}-4&1&-4&0&\color{orangered}{ 0 }&8&13\\& & -4& 32& \color{orangered}{-128} & & \\ \hline &1&-8&32&\color{orangered}{-128}&& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -4 } \cdot \color{blue}{ \left( -128 \right) } = \color{blue}{ 512 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{-4}&1&-4&0&0&8&13\\& & -4& 32& -128& \color{blue}{512} & \\ \hline &1&-8&32&\color{blue}{-128}&& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ 8 } + \color{orangered}{ 512 } = \color{orangered}{ 520 } $
$$ \begin{array}{c|rrrrrr}-4&1&-4&0&0&\color{orangered}{ 8 }&13\\& & -4& 32& -128& \color{orangered}{512} & \\ \hline &1&-8&32&-128&\color{orangered}{520}& \end{array} $$Step 10 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -4 } \cdot \color{blue}{ 520 } = \color{blue}{ -2080 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{-4}&1&-4&0&0&8&13\\& & -4& 32& -128& 512& \color{blue}{-2080} \\ \hline &1&-8&32&-128&\color{blue}{520}& \end{array} $$Step 11 : Add down last column: $ \color{orangered}{ 13 } + \color{orangered}{ \left( -2080 \right) } = \color{orangered}{ -2067 } $
$$ \begin{array}{c|rrrrrr}-4&1&-4&0&0&8&\color{orangered}{ 13 }\\& & -4& 32& -128& 512& \color{orangered}{-2080} \\ \hline &\color{blue}{1}&\color{blue}{-8}&\color{blue}{32}&\color{blue}{-128}&\color{blue}{520}&\color{orangered}{-2067} \end{array} $$Bottom line represents the quotient $ \color{blue}{ x^{4}-8x^{3}+32x^{2}-128x+520 } $ with a remainder of $ \color{red}{ -2067 } $.