Divide using synthetic division $$ ( x^{6}+4x^{4}-x^{3}+5 ) \div ( x+2 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrrrr}-2&1&0&4&-1&0&0&5\\& & -2& 4& -16& 34& -68& \color{black}{136} \\ \hline &\color{blue}{1}&\color{blue}{-2}&\color{blue}{8}&\color{blue}{-17}&\color{blue}{34}&\color{blue}{-68}&\color{orangered}{141} \end{array} $$The solution is:
$$ \frac{ x^{6}+4x^{4}-x^{3}+5 }{ x+2 } = \color{blue}{x^{5}-2x^{4}+8x^{3}-17x^{2}+34x-68} ~+~ \frac{ \color{red}{ 141 } }{ x+2 } $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x + 2 = 0 $ ( $ x = \color{blue}{ -2 } $ ) at the left.
$$ \begin{array}{c|rrrrrrr}\color{blue}{-2}&1&0&4&-1&0&0&5\\& & & & & & & \\ \hline &&&&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrrrr}-2&\color{orangered}{ 1 }&0&4&-1&0&0&5\\& & & & & & & \\ \hline &\color{orangered}{1}&&&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ 1 } = \color{blue}{ -2 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{-2}&1&0&4&-1&0&0&5\\& & \color{blue}{-2} & & & & & \\ \hline &\color{blue}{1}&&&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ \left( -2 \right) } = \color{orangered}{ -2 } $
$$ \begin{array}{c|rrrrrrr}-2&1&\color{orangered}{ 0 }&4&-1&0&0&5\\& & \color{orangered}{-2} & & & & & \\ \hline &1&\color{orangered}{-2}&&&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ \left( -2 \right) } = \color{blue}{ 4 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{-2}&1&0&4&-1&0&0&5\\& & -2& \color{blue}{4} & & & & \\ \hline &1&\color{blue}{-2}&&&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 4 } + \color{orangered}{ 4 } = \color{orangered}{ 8 } $
$$ \begin{array}{c|rrrrrrr}-2&1&0&\color{orangered}{ 4 }&-1&0&0&5\\& & -2& \color{orangered}{4} & & & & \\ \hline &1&-2&\color{orangered}{8}&&&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ 8 } = \color{blue}{ -16 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{-2}&1&0&4&-1&0&0&5\\& & -2& 4& \color{blue}{-16} & & & \\ \hline &1&-2&\color{blue}{8}&&&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -1 } + \color{orangered}{ \left( -16 \right) } = \color{orangered}{ -17 } $
$$ \begin{array}{c|rrrrrrr}-2&1&0&4&\color{orangered}{ -1 }&0&0&5\\& & -2& 4& \color{orangered}{-16} & & & \\ \hline &1&-2&8&\color{orangered}{-17}&&& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ \left( -17 \right) } = \color{blue}{ 34 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{-2}&1&0&4&-1&0&0&5\\& & -2& 4& -16& \color{blue}{34} & & \\ \hline &1&-2&8&\color{blue}{-17}&&& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 34 } = \color{orangered}{ 34 } $
$$ \begin{array}{c|rrrrrrr}-2&1&0&4&-1&\color{orangered}{ 0 }&0&5\\& & -2& 4& -16& \color{orangered}{34} & & \\ \hline &1&-2&8&-17&\color{orangered}{34}&& \end{array} $$Step 10 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ 34 } = \color{blue}{ -68 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{-2}&1&0&4&-1&0&0&5\\& & -2& 4& -16& 34& \color{blue}{-68} & \\ \hline &1&-2&8&-17&\color{blue}{34}&& \end{array} $$Step 11 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ \left( -68 \right) } = \color{orangered}{ -68 } $
$$ \begin{array}{c|rrrrrrr}-2&1&0&4&-1&0&\color{orangered}{ 0 }&5\\& & -2& 4& -16& 34& \color{orangered}{-68} & \\ \hline &1&-2&8&-17&34&\color{orangered}{-68}& \end{array} $$Step 12 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ \left( -68 \right) } = \color{blue}{ 136 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{-2}&1&0&4&-1&0&0&5\\& & -2& 4& -16& 34& -68& \color{blue}{136} \\ \hline &1&-2&8&-17&34&\color{blue}{-68}& \end{array} $$Step 13 : Add down last column: $ \color{orangered}{ 5 } + \color{orangered}{ 136 } = \color{orangered}{ 141 } $
$$ \begin{array}{c|rrrrrrr}-2&1&0&4&-1&0&0&\color{orangered}{ 5 }\\& & -2& 4& -16& 34& -68& \color{orangered}{136} \\ \hline &\color{blue}{1}&\color{blue}{-2}&\color{blue}{8}&\color{blue}{-17}&\color{blue}{34}&\color{blue}{-68}&\color{orangered}{141} \end{array} $$Bottom line represents the quotient $ \color{blue}{ x^{5}-2x^{4}+8x^{3}-17x^{2}+34x-68 } $ with a remainder of $ \color{red}{ 141 } $.