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