Divide using synthetic division $$ ( -9x^{5}-18x^{4}+4x^{3}+21x^{2}+10x ) \div ( -3x-5 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrrr}5&-9&-18&4&21&10&0\\& & -45& -315& -1555& -7670& \color{black}{-38300} \\ \hline &\color{blue}{-9}&\color{blue}{-63}&\color{blue}{-311}&\color{blue}{-1534}&\color{blue}{-7660}&\color{orangered}{-38300} \end{array} $$The solution is:
$$ \frac{ -9x^{5}-18x^{4}+4x^{3}+21x^{2}+10x }{ x-5 } = \color{blue}{-9x^{4}-63x^{3}-311x^{2}-1534x-7660} \color{red}{~-~} \frac{ \color{red}{ 38300 } }{ x-5 } $$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.
$$ \begin{array}{c|rrrrrr}\color{blue}{5}&-9&-18&4&21&10&0\\& & & & & & \\ \hline &&&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrrr}5&\color{orangered}{ -9 }&-18&4&21&10&0\\& & & & & & \\ \hline &\color{orangered}{-9}&&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 5 } \cdot \color{blue}{ \left( -9 \right) } = \color{blue}{ -45 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{5}&-9&-18&4&21&10&0\\& & \color{blue}{-45} & & & & \\ \hline &\color{blue}{-9}&&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -18 } + \color{orangered}{ \left( -45 \right) } = \color{orangered}{ -63 } $
$$ \begin{array}{c|rrrrrr}5&-9&\color{orangered}{ -18 }&4&21&10&0\\& & \color{orangered}{-45} & & & & \\ \hline &-9&\color{orangered}{-63}&&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 5 } \cdot \color{blue}{ \left( -63 \right) } = \color{blue}{ -315 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{5}&-9&-18&4&21&10&0\\& & -45& \color{blue}{-315} & & & \\ \hline &-9&\color{blue}{-63}&&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 4 } + \color{orangered}{ \left( -315 \right) } = \color{orangered}{ -311 } $
$$ \begin{array}{c|rrrrrr}5&-9&-18&\color{orangered}{ 4 }&21&10&0\\& & -45& \color{orangered}{-315} & & & \\ \hline &-9&-63&\color{orangered}{-311}&&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 5 } \cdot \color{blue}{ \left( -311 \right) } = \color{blue}{ -1555 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{5}&-9&-18&4&21&10&0\\& & -45& -315& \color{blue}{-1555} & & \\ \hline &-9&-63&\color{blue}{-311}&&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 21 } + \color{orangered}{ \left( -1555 \right) } = \color{orangered}{ -1534 } $
$$ \begin{array}{c|rrrrrr}5&-9&-18&4&\color{orangered}{ 21 }&10&0\\& & -45& -315& \color{orangered}{-1555} & & \\ \hline &-9&-63&-311&\color{orangered}{-1534}&& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 5 } \cdot \color{blue}{ \left( -1534 \right) } = \color{blue}{ -7670 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{5}&-9&-18&4&21&10&0\\& & -45& -315& -1555& \color{blue}{-7670} & \\ \hline &-9&-63&-311&\color{blue}{-1534}&& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ 10 } + \color{orangered}{ \left( -7670 \right) } = \color{orangered}{ -7660 } $
$$ \begin{array}{c|rrrrrr}5&-9&-18&4&21&\color{orangered}{ 10 }&0\\& & -45& -315& -1555& \color{orangered}{-7670} & \\ \hline &-9&-63&-311&-1534&\color{orangered}{-7660}& \end{array} $$Step 10 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 5 } \cdot \color{blue}{ \left( -7660 \right) } = \color{blue}{ -38300 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{5}&-9&-18&4&21&10&0\\& & -45& -315& -1555& -7670& \color{blue}{-38300} \\ \hline &-9&-63&-311&-1534&\color{blue}{-7660}& \end{array} $$Step 11 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ \left( -38300 \right) } = \color{orangered}{ -38300 } $
$$ \begin{array}{c|rrrrrr}5&-9&-18&4&21&10&\color{orangered}{ 0 }\\& & -45& -315& -1555& -7670& \color{orangered}{-38300} \\ \hline &\color{blue}{-9}&\color{blue}{-63}&\color{blue}{-311}&\color{blue}{-1534}&\color{blue}{-7660}&\color{orangered}{-38300} \end{array} $$Bottom line represents the quotient $ \color{blue}{ -9x^{4}-63x^{3}-311x^{2}-1534x-7660 } $ with a remainder of $ \color{red}{ -38300 } $.