Divide using synthetic division $$ ( -3x^{7}-x^{6}-5x^{3}+2x-8 ) \div ( 3x+5 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrrrrr}-5&-3&-1&0&0&-5&0&2&-8\\& & 15& -70& 350& -1750& 8775& -43875& \color{black}{219365} \\ \hline &\color{blue}{-3}&\color{blue}{14}&\color{blue}{-70}&\color{blue}{350}&\color{blue}{-1755}&\color{blue}{8775}&\color{blue}{-43873}&\color{orangered}{219357} \end{array} $$The solution is:
$$ \frac{ -3x^{7}-x^{6}-5x^{3}+2x-8 }{ x+5 } = \color{blue}{-3x^{6}+14x^{5}-70x^{4}+350x^{3}-1755x^{2}+8775x-43873} ~+~ \frac{ \color{red}{ 219357 } }{ x+5 } $$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.
$$ \begin{array}{c|rrrrrrrr}\color{blue}{-5}&-3&-1&0&0&-5&0&2&-8\\& & & & & & & & \\ \hline &&&&&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrrrrr}-5&\color{orangered}{ -3 }&-1&0&0&-5&0&2&-8\\& & & & & & & & \\ \hline &\color{orangered}{-3}&&&&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -5 } \cdot \color{blue}{ \left( -3 \right) } = \color{blue}{ 15 } $.
$$ \begin{array}{c|rrrrrrrr}\color{blue}{-5}&-3&-1&0&0&-5&0&2&-8\\& & \color{blue}{15} & & & & & & \\ \hline &\color{blue}{-3}&&&&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -1 } + \color{orangered}{ 15 } = \color{orangered}{ 14 } $
$$ \begin{array}{c|rrrrrrrr}-5&-3&\color{orangered}{ -1 }&0&0&-5&0&2&-8\\& & \color{orangered}{15} & & & & & & \\ \hline &-3&\color{orangered}{14}&&&&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -5 } \cdot \color{blue}{ 14 } = \color{blue}{ -70 } $.
$$ \begin{array}{c|rrrrrrrr}\color{blue}{-5}&-3&-1&0&0&-5&0&2&-8\\& & 15& \color{blue}{-70} & & & & & \\ \hline &-3&\color{blue}{14}&&&&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ \left( -70 \right) } = \color{orangered}{ -70 } $
$$ \begin{array}{c|rrrrrrrr}-5&-3&-1&\color{orangered}{ 0 }&0&-5&0&2&-8\\& & 15& \color{orangered}{-70} & & & & & \\ \hline &-3&14&\color{orangered}{-70}&&&&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -5 } \cdot \color{blue}{ \left( -70 \right) } = \color{blue}{ 350 } $.
$$ \begin{array}{c|rrrrrrrr}\color{blue}{-5}&-3&-1&0&0&-5&0&2&-8\\& & 15& -70& \color{blue}{350} & & & & \\ \hline &-3&14&\color{blue}{-70}&&&&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 350 } = \color{orangered}{ 350 } $
$$ \begin{array}{c|rrrrrrrr}-5&-3&-1&0&\color{orangered}{ 0 }&-5&0&2&-8\\& & 15& -70& \color{orangered}{350} & & & & \\ \hline &-3&14&-70&\color{orangered}{350}&&&& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -5 } \cdot \color{blue}{ 350 } = \color{blue}{ -1750 } $.
$$ \begin{array}{c|rrrrrrrr}\color{blue}{-5}&-3&-1&0&0&-5&0&2&-8\\& & 15& -70& 350& \color{blue}{-1750} & & & \\ \hline &-3&14&-70&\color{blue}{350}&&&& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ -5 } + \color{orangered}{ \left( -1750 \right) } = \color{orangered}{ -1755 } $
$$ \begin{array}{c|rrrrrrrr}-5&-3&-1&0&0&\color{orangered}{ -5 }&0&2&-8\\& & 15& -70& 350& \color{orangered}{-1750} & & & \\ \hline &-3&14&-70&350&\color{orangered}{-1755}&&& \end{array} $$Step 10 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -5 } \cdot \color{blue}{ \left( -1755 \right) } = \color{blue}{ 8775 } $.
$$ \begin{array}{c|rrrrrrrr}\color{blue}{-5}&-3&-1&0&0&-5&0&2&-8\\& & 15& -70& 350& -1750& \color{blue}{8775} & & \\ \hline &-3&14&-70&350&\color{blue}{-1755}&&& \end{array} $$Step 11 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 8775 } = \color{orangered}{ 8775 } $
$$ \begin{array}{c|rrrrrrrr}-5&-3&-1&0&0&-5&\color{orangered}{ 0 }&2&-8\\& & 15& -70& 350& -1750& \color{orangered}{8775} & & \\ \hline &-3&14&-70&350&-1755&\color{orangered}{8775}&& \end{array} $$Step 12 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -5 } \cdot \color{blue}{ 8775 } = \color{blue}{ -43875 } $.
$$ \begin{array}{c|rrrrrrrr}\color{blue}{-5}&-3&-1&0&0&-5&0&2&-8\\& & 15& -70& 350& -1750& 8775& \color{blue}{-43875} & \\ \hline &-3&14&-70&350&-1755&\color{blue}{8775}&& \end{array} $$Step 13 : Add down last column: $ \color{orangered}{ 2 } + \color{orangered}{ \left( -43875 \right) } = \color{orangered}{ -43873 } $
$$ \begin{array}{c|rrrrrrrr}-5&-3&-1&0&0&-5&0&\color{orangered}{ 2 }&-8\\& & 15& -70& 350& -1750& 8775& \color{orangered}{-43875} & \\ \hline &-3&14&-70&350&-1755&8775&\color{orangered}{-43873}& \end{array} $$Step 14 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -5 } \cdot \color{blue}{ \left( -43873 \right) } = \color{blue}{ 219365 } $.
$$ \begin{array}{c|rrrrrrrr}\color{blue}{-5}&-3&-1&0&0&-5&0&2&-8\\& & 15& -70& 350& -1750& 8775& -43875& \color{blue}{219365} \\ \hline &-3&14&-70&350&-1755&8775&\color{blue}{-43873}& \end{array} $$Step 15 : Add down last column: $ \color{orangered}{ -8 } + \color{orangered}{ 219365 } = \color{orangered}{ 219357 } $
$$ \begin{array}{c|rrrrrrrr}-5&-3&-1&0&0&-5&0&2&\color{orangered}{ -8 }\\& & 15& -70& 350& -1750& 8775& -43875& \color{orangered}{219365} \\ \hline &\color{blue}{-3}&\color{blue}{14}&\color{blue}{-70}&\color{blue}{350}&\color{blue}{-1755}&\color{blue}{8775}&\color{blue}{-43873}&\color{orangered}{219357} \end{array} $$Bottom line represents the quotient $ \color{blue}{ -3x^{6}+14x^{5}-70x^{4}+350x^{3}-1755x^{2}+8775x-43873 } $ with a remainder of $ \color{red}{ 219357 } $.