Divide using synthetic division $$ ( 8x^{6}+40x^{5}+4x^{4}+30x^{3}+60x^{2}+45x-26 ) \div ( x+5 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrrrr}-5&8&40&4&30&60&45&-26\\& & -40& 0& -20& -50& -50& \color{black}{25} \\ \hline &\color{blue}{8}&\color{blue}{0}&\color{blue}{4}&\color{blue}{10}&\color{blue}{10}&\color{blue}{-5}&\color{orangered}{-1} \end{array} $$The solution is:
$$ \frac{ 8x^{6}+40x^{5}+4x^{4}+30x^{3}+60x^{2}+45x-26 }{ x+5 } = \color{blue}{8x^{5}+4x^{3}+10x^{2}+10x-5} \color{red}{~-~} \frac{ \color{red}{ 1 } }{ x+5 } $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x + 5 = 0 $ ( $ x = \color{blue}{ -5 } $ ) at the left.
$$ \begin{array}{c|rrrrrrr}\color{blue}{-5}&8&40&4&30&60&45&-26\\& & & & & & & \\ \hline &&&&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrrrr}-5&\color{orangered}{ 8 }&40&4&30&60&45&-26\\& & & & & & & \\ \hline &\color{orangered}{8}&&&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -5 } \cdot \color{blue}{ 8 } = \color{blue}{ -40 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{-5}&8&40&4&30&60&45&-26\\& & \color{blue}{-40} & & & & & \\ \hline &\color{blue}{8}&&&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 40 } + \color{orangered}{ \left( -40 \right) } = \color{orangered}{ 0 } $
$$ \begin{array}{c|rrrrrrr}-5&8&\color{orangered}{ 40 }&4&30&60&45&-26\\& & \color{orangered}{-40} & & & & & \\ \hline &8&\color{orangered}{0}&&&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -5 } \cdot \color{blue}{ 0 } = \color{blue}{ 0 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{-5}&8&40&4&30&60&45&-26\\& & -40& \color{blue}{0} & & & & \\ \hline &8&\color{blue}{0}&&&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 4 } + \color{orangered}{ 0 } = \color{orangered}{ 4 } $
$$ \begin{array}{c|rrrrrrr}-5&8&40&\color{orangered}{ 4 }&30&60&45&-26\\& & -40& \color{orangered}{0} & & & & \\ \hline &8&0&\color{orangered}{4}&&&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -5 } \cdot \color{blue}{ 4 } = \color{blue}{ -20 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{-5}&8&40&4&30&60&45&-26\\& & -40& 0& \color{blue}{-20} & & & \\ \hline &8&0&\color{blue}{4}&&&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 30 } + \color{orangered}{ \left( -20 \right) } = \color{orangered}{ 10 } $
$$ \begin{array}{c|rrrrrrr}-5&8&40&4&\color{orangered}{ 30 }&60&45&-26\\& & -40& 0& \color{orangered}{-20} & & & \\ \hline &8&0&4&\color{orangered}{10}&&& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -5 } \cdot \color{blue}{ 10 } = \color{blue}{ -50 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{-5}&8&40&4&30&60&45&-26\\& & -40& 0& -20& \color{blue}{-50} & & \\ \hline &8&0&4&\color{blue}{10}&&& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ 60 } + \color{orangered}{ \left( -50 \right) } = \color{orangered}{ 10 } $
$$ \begin{array}{c|rrrrrrr}-5&8&40&4&30&\color{orangered}{ 60 }&45&-26\\& & -40& 0& -20& \color{orangered}{-50} & & \\ \hline &8&0&4&10&\color{orangered}{10}&& \end{array} $$Step 10 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -5 } \cdot \color{blue}{ 10 } = \color{blue}{ -50 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{-5}&8&40&4&30&60&45&-26\\& & -40& 0& -20& -50& \color{blue}{-50} & \\ \hline &8&0&4&10&\color{blue}{10}&& \end{array} $$Step 11 : Add down last column: $ \color{orangered}{ 45 } + \color{orangered}{ \left( -50 \right) } = \color{orangered}{ -5 } $
$$ \begin{array}{c|rrrrrrr}-5&8&40&4&30&60&\color{orangered}{ 45 }&-26\\& & -40& 0& -20& -50& \color{orangered}{-50} & \\ \hline &8&0&4&10&10&\color{orangered}{-5}& \end{array} $$Step 12 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -5 } \cdot \color{blue}{ \left( -5 \right) } = \color{blue}{ 25 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{-5}&8&40&4&30&60&45&-26\\& & -40& 0& -20& -50& -50& \color{blue}{25} \\ \hline &8&0&4&10&10&\color{blue}{-5}& \end{array} $$Step 13 : Add down last column: $ \color{orangered}{ -26 } + \color{orangered}{ 25 } = \color{orangered}{ -1 } $
$$ \begin{array}{c|rrrrrrr}-5&8&40&4&30&60&45&\color{orangered}{ -26 }\\& & -40& 0& -20& -50& -50& \color{orangered}{25} \\ \hline &\color{blue}{8}&\color{blue}{0}&\color{blue}{4}&\color{blue}{10}&\color{blue}{10}&\color{blue}{-5}&\color{orangered}{-1} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 8x^{5}+4x^{3}+10x^{2}+10x-5 } $ with a remainder of $ \color{red}{ -1 } $.