Divide using synthetic division $$ ( 10x^{3}+x^{2}-21x+9 ) \div ( 5x-7 ) $$
The synthetic division table is:
$$ \begin{array}{c|rrrr}7&10&1&-21&9\\& & 70& 497& \color{black}{3332} \\ \hline &\color{blue}{10}&\color{blue}{71}&\color{blue}{476}&\color{orangered}{3341} \end{array} $$The solution is:
$$ \dfrac{ 10x^{3}+x^{2}-21x+9 }{ x-7 } = \color{blue}{10x^{2}+71x+476} ~+~ \dfrac{ \color{red}{ 3341 } }{ x-7 } $$Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x -7 = 0 $ ( $ x = \color{blue}{ 7 } $ ) at the left.
$$ \begin{array}{c|rrrr}\color{blue}{7}&10&1&-21&9\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}7&\color{orangered}{ 10 }&1&-21&9\\& & & & \\ \hline &\color{orangered}{10}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 7 } \cdot \color{blue}{ 10 } = \color{blue}{ 70 } $.
$$ \begin{array}{c|rrrr}\color{blue}{7}&10&1&-21&9\\& & \color{blue}{70} & & \\ \hline &\color{blue}{10}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 1 } + \color{orangered}{ 70 } = \color{orangered}{ 71 } $
$$ \begin{array}{c|rrrr}7&10&\color{orangered}{ 1 }&-21&9\\& & \color{orangered}{70} & & \\ \hline &10&\color{orangered}{71}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 7 } \cdot \color{blue}{ 71 } = \color{blue}{ 497 } $.
$$ \begin{array}{c|rrrr}\color{blue}{7}&10&1&-21&9\\& & 70& \color{blue}{497} & \\ \hline &10&\color{blue}{71}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -21 } + \color{orangered}{ 497 } = \color{orangered}{ 476 } $
$$ \begin{array}{c|rrrr}7&10&1&\color{orangered}{ -21 }&9\\& & 70& \color{orangered}{497} & \\ \hline &10&71&\color{orangered}{476}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 7 } \cdot \color{blue}{ 476 } = \color{blue}{ 3332 } $.
$$ \begin{array}{c|rrrr}\color{blue}{7}&10&1&-21&9\\& & 70& 497& \color{blue}{3332} \\ \hline &10&71&\color{blue}{476}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 9 } + \color{orangered}{ 3332 } = \color{orangered}{ 3341 } $
$$ \begin{array}{c|rrrr}7&10&1&-21&\color{orangered}{ 9 }\\& & 70& 497& \color{orangered}{3332} \\ \hline &\color{blue}{10}&\color{blue}{71}&\color{blue}{476}&\color{orangered}{3341} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 10x^{2}+71x+476 } $ with a remainder of $ \color{red}{ 3341 } $.