Divide using synthetic division $$ ( 9x^{4}+37x^{3}+29x^{2}+4x-21 ) \div ( x-2 ) $$
The synthetic division table is:
$$ \begin{array}{c|rrrrr}2&9&37&29&4&-21\\& & 18& 110& 278& \color{black}{564} \\ \hline &\color{blue}{9}&\color{blue}{55}&\color{blue}{139}&\color{blue}{282}&\color{orangered}{543} \end{array} $$The solution is:
$$ \dfrac{ 9x^{4}+37x^{3}+29x^{2}+4x-21 }{ x-2 } = \color{blue}{9x^{3}+55x^{2}+139x+282} ~+~ \dfrac{ \color{red}{ 543 } }{ x-2 } $$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.
$$ \begin{array}{c|rrrrr}\color{blue}{2}&9&37&29&4&-21\\& & & & & \\ \hline &&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrr}2&\color{orangered}{ 9 }&37&29&4&-21\\& & & & & \\ \hline &\color{orangered}{9}&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ 9 } = \color{blue}{ 18 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{2}&9&37&29&4&-21\\& & \color{blue}{18} & & & \\ \hline &\color{blue}{9}&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 37 } + \color{orangered}{ 18 } = \color{orangered}{ 55 } $
$$ \begin{array}{c|rrrrr}2&9&\color{orangered}{ 37 }&29&4&-21\\& & \color{orangered}{18} & & & \\ \hline &9&\color{orangered}{55}&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ 55 } = \color{blue}{ 110 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{2}&9&37&29&4&-21\\& & 18& \color{blue}{110} & & \\ \hline &9&\color{blue}{55}&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 29 } + \color{orangered}{ 110 } = \color{orangered}{ 139 } $
$$ \begin{array}{c|rrrrr}2&9&37&\color{orangered}{ 29 }&4&-21\\& & 18& \color{orangered}{110} & & \\ \hline &9&55&\color{orangered}{139}&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ 139 } = \color{blue}{ 278 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{2}&9&37&29&4&-21\\& & 18& 110& \color{blue}{278} & \\ \hline &9&55&\color{blue}{139}&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 4 } + \color{orangered}{ 278 } = \color{orangered}{ 282 } $
$$ \begin{array}{c|rrrrr}2&9&37&29&\color{orangered}{ 4 }&-21\\& & 18& 110& \color{orangered}{278} & \\ \hline &9&55&139&\color{orangered}{282}& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ 282 } = \color{blue}{ 564 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{2}&9&37&29&4&-21\\& & 18& 110& 278& \color{blue}{564} \\ \hline &9&55&139&\color{blue}{282}& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ -21 } + \color{orangered}{ 564 } = \color{orangered}{ 543 } $
$$ \begin{array}{c|rrrrr}2&9&37&29&4&\color{orangered}{ -21 }\\& & 18& 110& 278& \color{orangered}{564} \\ \hline &\color{blue}{9}&\color{blue}{55}&\color{blue}{139}&\color{blue}{282}&\color{orangered}{543} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 9x^{3}+55x^{2}+139x+282 } $ with a remainder of $ \color{red}{ 543 } $.