Divide using synthetic division $$ ( 8x^{8}+x^{5}+17x^{3}+8x^{2}+12x-17 ) \div ( x+4 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrrrrrr}-4&8&0&0&1&0&17&8&12&-17\\& & -32& 128& -512& 2044& -8176& 32636& -130576& \color{black}{522256} \\ \hline &\color{blue}{8}&\color{blue}{-32}&\color{blue}{128}&\color{blue}{-511}&\color{blue}{2044}&\color{blue}{-8159}&\color{blue}{32644}&\color{blue}{-130564}&\color{orangered}{522239} \end{array} $$The solution is:
$$ \frac{ 8x^{8}+x^{5}+17x^{3}+8x^{2}+12x-17 }{ x+4 } = \color{blue}{8x^{7}-32x^{6}+128x^{5}-511x^{4}+2044x^{3}-8159x^{2}+32644x-130564} ~+~ \frac{ \color{red}{ 522239 } }{ x+4 } $$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.
$$ \begin{array}{c|rrrrrrrrr}\color{blue}{-4}&8&0&0&1&0&17&8&12&-17\\& & & & & & & & & \\ \hline &&&&&&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrrrrrr}-4&\color{orangered}{ 8 }&0&0&1&0&17&8&12&-17\\& & & & & & & & & \\ \hline &\color{orangered}{8}&&&&&&&& \end{array} $$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 } $.
$$ \begin{array}{c|rrrrrrrrr}\color{blue}{-4}&8&0&0&1&0&17&8&12&-17\\& & \color{blue}{-32} & & & & & & & \\ \hline &\color{blue}{8}&&&&&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ \left( -32 \right) } = \color{orangered}{ -32 } $
$$ \begin{array}{c|rrrrrrrrr}-4&8&\color{orangered}{ 0 }&0&1&0&17&8&12&-17\\& & \color{orangered}{-32} & & & & & & & \\ \hline &8&\color{orangered}{-32}&&&&&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -4 } \cdot \color{blue}{ \left( -32 \right) } = \color{blue}{ 128 } $.
$$ \begin{array}{c|rrrrrrrrr}\color{blue}{-4}&8&0&0&1&0&17&8&12&-17\\& & -32& \color{blue}{128} & & & & & & \\ \hline &8&\color{blue}{-32}&&&&&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 128 } = \color{orangered}{ 128 } $
$$ \begin{array}{c|rrrrrrrrr}-4&8&0&\color{orangered}{ 0 }&1&0&17&8&12&-17\\& & -32& \color{orangered}{128} & & & & & & \\ \hline &8&-32&\color{orangered}{128}&&&&&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -4 } \cdot \color{blue}{ 128 } = \color{blue}{ -512 } $.
$$ \begin{array}{c|rrrrrrrrr}\color{blue}{-4}&8&0&0&1&0&17&8&12&-17\\& & -32& 128& \color{blue}{-512} & & & & & \\ \hline &8&-32&\color{blue}{128}&&&&&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 1 } + \color{orangered}{ \left( -512 \right) } = \color{orangered}{ -511 } $
$$ \begin{array}{c|rrrrrrrrr}-4&8&0&0&\color{orangered}{ 1 }&0&17&8&12&-17\\& & -32& 128& \color{orangered}{-512} & & & & & \\ \hline &8&-32&128&\color{orangered}{-511}&&&&& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -4 } \cdot \color{blue}{ \left( -511 \right) } = \color{blue}{ 2044 } $.
$$ \begin{array}{c|rrrrrrrrr}\color{blue}{-4}&8&0&0&1&0&17&8&12&-17\\& & -32& 128& -512& \color{blue}{2044} & & & & \\ \hline &8&-32&128&\color{blue}{-511}&&&&& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 2044 } = \color{orangered}{ 2044 } $
$$ \begin{array}{c|rrrrrrrrr}-4&8&0&0&1&\color{orangered}{ 0 }&17&8&12&-17\\& & -32& 128& -512& \color{orangered}{2044} & & & & \\ \hline &8&-32&128&-511&\color{orangered}{2044}&&&& \end{array} $$Step 10 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -4 } \cdot \color{blue}{ 2044 } = \color{blue}{ -8176 } $.
$$ \begin{array}{c|rrrrrrrrr}\color{blue}{-4}&8&0&0&1&0&17&8&12&-17\\& & -32& 128& -512& 2044& \color{blue}{-8176} & & & \\ \hline &8&-32&128&-511&\color{blue}{2044}&&&& \end{array} $$Step 11 : Add down last column: $ \color{orangered}{ 17 } + \color{orangered}{ \left( -8176 \right) } = \color{orangered}{ -8159 } $
$$ \begin{array}{c|rrrrrrrrr}-4&8&0&0&1&0&\color{orangered}{ 17 }&8&12&-17\\& & -32& 128& -512& 2044& \color{orangered}{-8176} & & & \\ \hline &8&-32&128&-511&2044&\color{orangered}{-8159}&&& \end{array} $$Step 12 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -4 } \cdot \color{blue}{ \left( -8159 \right) } = \color{blue}{ 32636 } $.
$$ \begin{array}{c|rrrrrrrrr}\color{blue}{-4}&8&0&0&1&0&17&8&12&-17\\& & -32& 128& -512& 2044& -8176& \color{blue}{32636} & & \\ \hline &8&-32&128&-511&2044&\color{blue}{-8159}&&& \end{array} $$Step 13 : Add down last column: $ \color{orangered}{ 8 } + \color{orangered}{ 32636 } = \color{orangered}{ 32644 } $
$$ \begin{array}{c|rrrrrrrrr}-4&8&0&0&1&0&17&\color{orangered}{ 8 }&12&-17\\& & -32& 128& -512& 2044& -8176& \color{orangered}{32636} & & \\ \hline &8&-32&128&-511&2044&-8159&\color{orangered}{32644}&& \end{array} $$Step 14 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -4 } \cdot \color{blue}{ 32644 } = \color{blue}{ -130576 } $.
$$ \begin{array}{c|rrrrrrrrr}\color{blue}{-4}&8&0&0&1&0&17&8&12&-17\\& & -32& 128& -512& 2044& -8176& 32636& \color{blue}{-130576} & \\ \hline &8&-32&128&-511&2044&-8159&\color{blue}{32644}&& \end{array} $$Step 15 : Add down last column: $ \color{orangered}{ 12 } + \color{orangered}{ \left( -130576 \right) } = \color{orangered}{ -130564 } $
$$ \begin{array}{c|rrrrrrrrr}-4&8&0&0&1&0&17&8&\color{orangered}{ 12 }&-17\\& & -32& 128& -512& 2044& -8176& 32636& \color{orangered}{-130576} & \\ \hline &8&-32&128&-511&2044&-8159&32644&\color{orangered}{-130564}& \end{array} $$Step 16 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -4 } \cdot \color{blue}{ \left( -130564 \right) } = \color{blue}{ 522256 } $.
$$ \begin{array}{c|rrrrrrrrr}\color{blue}{-4}&8&0&0&1&0&17&8&12&-17\\& & -32& 128& -512& 2044& -8176& 32636& -130576& \color{blue}{522256} \\ \hline &8&-32&128&-511&2044&-8159&32644&\color{blue}{-130564}& \end{array} $$Step 17 : Add down last column: $ \color{orangered}{ -17 } + \color{orangered}{ 522256 } = \color{orangered}{ 522239 } $
$$ \begin{array}{c|rrrrrrrrr}-4&8&0&0&1&0&17&8&12&\color{orangered}{ -17 }\\& & -32& 128& -512& 2044& -8176& 32636& -130576& \color{orangered}{522256} \\ \hline &\color{blue}{8}&\color{blue}{-32}&\color{blue}{128}&\color{blue}{-511}&\color{blue}{2044}&\color{blue}{-8159}&\color{blue}{32644}&\color{blue}{-130564}&\color{orangered}{522239} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 8x^{7}-32x^{6}+128x^{5}-511x^{4}+2044x^{3}-8159x^{2}+32644x-130564 } $ with a remainder of $ \color{red}{ 522239 } $.