Divide using synthetic division $$ ( 8x^{5}+32x^{4}+5x+20 ) \div ( x+4 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrrr}-4&8&32&0&0&5&20\\& & -32& 0& 0& 0& \color{black}{-20} \\ \hline &\color{blue}{8}&\color{blue}{0}&\color{blue}{0}&\color{blue}{0}&\color{blue}{5}&\color{orangered}{0} \end{array} $$The solution is:
$$ \frac{ 8x^{5}+32x^{4}+5x+20 }{ x+4 } = \color{blue}{8x^{4}+5} $$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.
$$ \begin{array}{c|rrrrrr}\color{blue}{-4}&8&32&0&0&5&20\\& & & & & & \\ \hline &&&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrrr}-4&\color{orangered}{ 8 }&32&0&0&5&20\\& & & & & & \\ \hline &\color{orangered}{8}&&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -4 } \cdot \color{blue}{ 8 } = \color{blue}{ -32 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{-4}&8&32&0&0&5&20\\& & \color{blue}{-32} & & & & \\ \hline &\color{blue}{8}&&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 32 } + \color{orangered}{ \left( -32 \right) } = \color{orangered}{ 0 } $
$$ \begin{array}{c|rrrrrr}-4&8&\color{orangered}{ 32 }&0&0&5&20\\& & \color{orangered}{-32} & & & & \\ \hline &8&\color{orangered}{0}&&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -4 } \cdot \color{blue}{ 0 } = \color{blue}{ 0 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{-4}&8&32&0&0&5&20\\& & -32& \color{blue}{0} & & & \\ \hline &8&\color{blue}{0}&&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 0 } = \color{orangered}{ 0 } $
$$ \begin{array}{c|rrrrrr}-4&8&32&\color{orangered}{ 0 }&0&5&20\\& & -32& \color{orangered}{0} & & & \\ \hline &8&0&\color{orangered}{0}&&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -4 } \cdot \color{blue}{ 0 } = \color{blue}{ 0 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{-4}&8&32&0&0&5&20\\& & -32& 0& \color{blue}{0} & & \\ \hline &8&0&\color{blue}{0}&&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 0 } = \color{orangered}{ 0 } $
$$ \begin{array}{c|rrrrrr}-4&8&32&0&\color{orangered}{ 0 }&5&20\\& & -32& 0& \color{orangered}{0} & & \\ \hline &8&0&0&\color{orangered}{0}&& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -4 } \cdot \color{blue}{ 0 } = \color{blue}{ 0 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{-4}&8&32&0&0&5&20\\& & -32& 0& 0& \color{blue}{0} & \\ \hline &8&0&0&\color{blue}{0}&& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ 5 } + \color{orangered}{ 0 } = \color{orangered}{ 5 } $
$$ \begin{array}{c|rrrrrr}-4&8&32&0&0&\color{orangered}{ 5 }&20\\& & -32& 0& 0& \color{orangered}{0} & \\ \hline &8&0&0&0&\color{orangered}{5}& \end{array} $$Step 10 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -4 } \cdot \color{blue}{ 5 } = \color{blue}{ -20 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{-4}&8&32&0&0&5&20\\& & -32& 0& 0& 0& \color{blue}{-20} \\ \hline &8&0&0&0&\color{blue}{5}& \end{array} $$Step 11 : Add down last column: $ \color{orangered}{ 20 } + \color{orangered}{ \left( -20 \right) } = \color{orangered}{ 0 } $
$$ \begin{array}{c|rrrrrr}-4&8&32&0&0&5&\color{orangered}{ 20 }\\& & -32& 0& 0& 0& \color{orangered}{-20} \\ \hline &\color{blue}{8}&\color{blue}{0}&\color{blue}{0}&\color{blue}{0}&\color{blue}{5}&\color{orangered}{0} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 8x^{4}+5 } $ with a remainder of $ \color{red}{ 0 } $.