Divide using synthetic division $$ ( x^{5}+6x^{4}-3x^{2}-22x-29 ) \div ( x-6 ) $$
The synthetic division table is:
$$ \begin{array}{c|rrrrrr}6&1&6&0&-3&-22&-29\\& & 6& 72& 432& 2574& \color{black}{15312} \\ \hline &\color{blue}{1}&\color{blue}{12}&\color{blue}{72}&\color{blue}{429}&\color{blue}{2552}&\color{orangered}{15283} \end{array} $$The solution is:
$$ \dfrac{ x^{5}+6x^{4}-3x^{2}-22x-29 }{ x-6 } = \color{blue}{x^{4}+12x^{3}+72x^{2}+429x+2552} ~+~ \dfrac{ \color{red}{ 15283 } }{ x-6 } $$Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x -6 = 0 $ ( $ x = \color{blue}{ 6 } $ ) at the left.
$$ \begin{array}{c|rrrrrr}\color{blue}{6}&1&6&0&-3&-22&-29\\& & & & & & \\ \hline &&&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrrr}6&\color{orangered}{ 1 }&6&0&-3&-22&-29\\& & & & & & \\ \hline &\color{orangered}{1}&&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 6 } \cdot \color{blue}{ 1 } = \color{blue}{ 6 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{6}&1&6&0&-3&-22&-29\\& & \color{blue}{6} & & & & \\ \hline &\color{blue}{1}&&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 6 } + \color{orangered}{ 6 } = \color{orangered}{ 12 } $
$$ \begin{array}{c|rrrrrr}6&1&\color{orangered}{ 6 }&0&-3&-22&-29\\& & \color{orangered}{6} & & & & \\ \hline &1&\color{orangered}{12}&&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 6 } \cdot \color{blue}{ 12 } = \color{blue}{ 72 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{6}&1&6&0&-3&-22&-29\\& & 6& \color{blue}{72} & & & \\ \hline &1&\color{blue}{12}&&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 72 } = \color{orangered}{ 72 } $
$$ \begin{array}{c|rrrrrr}6&1&6&\color{orangered}{ 0 }&-3&-22&-29\\& & 6& \color{orangered}{72} & & & \\ \hline &1&12&\color{orangered}{72}&&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 6 } \cdot \color{blue}{ 72 } = \color{blue}{ 432 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{6}&1&6&0&-3&-22&-29\\& & 6& 72& \color{blue}{432} & & \\ \hline &1&12&\color{blue}{72}&&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -3 } + \color{orangered}{ 432 } = \color{orangered}{ 429 } $
$$ \begin{array}{c|rrrrrr}6&1&6&0&\color{orangered}{ -3 }&-22&-29\\& & 6& 72& \color{orangered}{432} & & \\ \hline &1&12&72&\color{orangered}{429}&& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 6 } \cdot \color{blue}{ 429 } = \color{blue}{ 2574 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{6}&1&6&0&-3&-22&-29\\& & 6& 72& 432& \color{blue}{2574} & \\ \hline &1&12&72&\color{blue}{429}&& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ -22 } + \color{orangered}{ 2574 } = \color{orangered}{ 2552 } $
$$ \begin{array}{c|rrrrrr}6&1&6&0&-3&\color{orangered}{ -22 }&-29\\& & 6& 72& 432& \color{orangered}{2574} & \\ \hline &1&12&72&429&\color{orangered}{2552}& \end{array} $$Step 10 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 6 } \cdot \color{blue}{ 2552 } = \color{blue}{ 15312 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{6}&1&6&0&-3&-22&-29\\& & 6& 72& 432& 2574& \color{blue}{15312} \\ \hline &1&12&72&429&\color{blue}{2552}& \end{array} $$Step 11 : Add down last column: $ \color{orangered}{ -29 } + \color{orangered}{ 15312 } = \color{orangered}{ 15283 } $
$$ \begin{array}{c|rrrrrr}6&1&6&0&-3&-22&\color{orangered}{ -29 }\\& & 6& 72& 432& 2574& \color{orangered}{15312} \\ \hline &\color{blue}{1}&\color{blue}{12}&\color{blue}{72}&\color{blue}{429}&\color{blue}{2552}&\color{orangered}{15283} \end{array} $$Bottom line represents the quotient $ \color{blue}{ x^{4}+12x^{3}+72x^{2}+429x+2552 } $ with a remainder of $ \color{red}{ 15283 } $.