Divide using synthetic division $$ ( x^{5}+8x^{4}+17x^{3}+8x^{2}+12x-17 ) \div ( x+4 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrrr}-4&1&8&17&8&12&-17\\& & -4& -16& -4& -16& \color{black}{16} \\ \hline &\color{blue}{1}&\color{blue}{4}&\color{blue}{1}&\color{blue}{4}&\color{blue}{-4}&\color{orangered}{-1} \end{array} $$The solution is:
$$ \frac{ x^{5}+8x^{4}+17x^{3}+8x^{2}+12x-17 }{ x+4 } = \color{blue}{x^{4}+4x^{3}+x^{2}+4x-4} \color{red}{~-~} \frac{ \color{red}{ 1 } }{ x+4 } $$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.
$$ \begin{array}{c|rrrrrr}\color{blue}{-4}&1&8&17&8&12&-17\\& & & & & & \\ \hline &&&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrrr}-4&\color{orangered}{ 1 }&8&17&8&12&-17\\& & & & & & \\ \hline &\color{orangered}{1}&&&&& \end{array} $$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 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{-4}&1&8&17&8&12&-17\\& & \color{blue}{-4} & & & & \\ \hline &\color{blue}{1}&&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 8 } + \color{orangered}{ \left( -4 \right) } = \color{orangered}{ 4 } $
$$ \begin{array}{c|rrrrrr}-4&1&\color{orangered}{ 8 }&17&8&12&-17\\& & \color{orangered}{-4} & & & & \\ \hline &1&\color{orangered}{4}&&&& \end{array} $$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 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{-4}&1&8&17&8&12&-17\\& & -4& \color{blue}{-16} & & & \\ \hline &1&\color{blue}{4}&&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 17 } + \color{orangered}{ \left( -16 \right) } = \color{orangered}{ 1 } $
$$ \begin{array}{c|rrrrrr}-4&1&8&\color{orangered}{ 17 }&8&12&-17\\& & -4& \color{orangered}{-16} & & & \\ \hline &1&4&\color{orangered}{1}&&& \end{array} $$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 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{-4}&1&8&17&8&12&-17\\& & -4& -16& \color{blue}{-4} & & \\ \hline &1&4&\color{blue}{1}&&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 8 } + \color{orangered}{ \left( -4 \right) } = \color{orangered}{ 4 } $
$$ \begin{array}{c|rrrrrr}-4&1&8&17&\color{orangered}{ 8 }&12&-17\\& & -4& -16& \color{orangered}{-4} & & \\ \hline &1&4&1&\color{orangered}{4}&& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -4 } \cdot \color{blue}{ 4 } = \color{blue}{ -16 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{-4}&1&8&17&8&12&-17\\& & -4& -16& -4& \color{blue}{-16} & \\ \hline &1&4&1&\color{blue}{4}&& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ 12 } + \color{orangered}{ \left( -16 \right) } = \color{orangered}{ -4 } $
$$ \begin{array}{c|rrrrrr}-4&1&8&17&8&\color{orangered}{ 12 }&-17\\& & -4& -16& -4& \color{orangered}{-16} & \\ \hline &1&4&1&4&\color{orangered}{-4}& \end{array} $$Step 10 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -4 } \cdot \color{blue}{ \left( -4 \right) } = \color{blue}{ 16 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{-4}&1&8&17&8&12&-17\\& & -4& -16& -4& -16& \color{blue}{16} \\ \hline &1&4&1&4&\color{blue}{-4}& \end{array} $$Step 11 : Add down last column: $ \color{orangered}{ -17 } + \color{orangered}{ 16 } = \color{orangered}{ -1 } $
$$ \begin{array}{c|rrrrrr}-4&1&8&17&8&12&\color{orangered}{ -17 }\\& & -4& -16& -4& -16& \color{orangered}{16} \\ \hline &\color{blue}{1}&\color{blue}{4}&\color{blue}{1}&\color{blue}{4}&\color{blue}{-4}&\color{orangered}{-1} \end{array} $$Bottom line represents the quotient $ \color{blue}{ x^{4}+4x^{3}+x^{2}+4x-4 } $ with a remainder of $ \color{red}{ -1 } $.