Divide using synthetic division $$ ( 6x^{5}+5x^{4}-29x^{3}-25x^{2}-5x ) \div ( x-1 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrrr}1&6&5&-29&-25&-5&0\\& & 6& 11& -18& -43& \color{black}{-48} \\ \hline &\color{blue}{6}&\color{blue}{11}&\color{blue}{-18}&\color{blue}{-43}&\color{blue}{-48}&\color{orangered}{-48} \end{array} $$The solution is:
$$ \frac{ 6x^{5}+5x^{4}-29x^{3}-25x^{2}-5x }{ x-1 } = \color{blue}{6x^{4}+11x^{3}-18x^{2}-43x-48} \color{red}{~-~} \frac{ \color{red}{ 48 } }{ x-1 } $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x -1 = 0 $ ( $ x = \color{blue}{ 1 } $ ) at the left.
$$ \begin{array}{c|rrrrrr}\color{blue}{1}&6&5&-29&-25&-5&0\\& & & & & & \\ \hline &&&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrrr}1&\color{orangered}{ 6 }&5&-29&-25&-5&0\\& & & & & & \\ \hline &\color{orangered}{6}&&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 1 } \cdot \color{blue}{ 6 } = \color{blue}{ 6 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{1}&6&5&-29&-25&-5&0\\& & \color{blue}{6} & & & & \\ \hline &\color{blue}{6}&&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 5 } + \color{orangered}{ 6 } = \color{orangered}{ 11 } $
$$ \begin{array}{c|rrrrrr}1&6&\color{orangered}{ 5 }&-29&-25&-5&0\\& & \color{orangered}{6} & & & & \\ \hline &6&\color{orangered}{11}&&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 1 } \cdot \color{blue}{ 11 } = \color{blue}{ 11 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{1}&6&5&-29&-25&-5&0\\& & 6& \color{blue}{11} & & & \\ \hline &6&\color{blue}{11}&&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -29 } + \color{orangered}{ 11 } = \color{orangered}{ -18 } $
$$ \begin{array}{c|rrrrrr}1&6&5&\color{orangered}{ -29 }&-25&-5&0\\& & 6& \color{orangered}{11} & & & \\ \hline &6&11&\color{orangered}{-18}&&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 1 } \cdot \color{blue}{ \left( -18 \right) } = \color{blue}{ -18 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{1}&6&5&-29&-25&-5&0\\& & 6& 11& \color{blue}{-18} & & \\ \hline &6&11&\color{blue}{-18}&&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -25 } + \color{orangered}{ \left( -18 \right) } = \color{orangered}{ -43 } $
$$ \begin{array}{c|rrrrrr}1&6&5&-29&\color{orangered}{ -25 }&-5&0\\& & 6& 11& \color{orangered}{-18} & & \\ \hline &6&11&-18&\color{orangered}{-43}&& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 1 } \cdot \color{blue}{ \left( -43 \right) } = \color{blue}{ -43 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{1}&6&5&-29&-25&-5&0\\& & 6& 11& -18& \color{blue}{-43} & \\ \hline &6&11&-18&\color{blue}{-43}&& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ -5 } + \color{orangered}{ \left( -43 \right) } = \color{orangered}{ -48 } $
$$ \begin{array}{c|rrrrrr}1&6&5&-29&-25&\color{orangered}{ -5 }&0\\& & 6& 11& -18& \color{orangered}{-43} & \\ \hline &6&11&-18&-43&\color{orangered}{-48}& \end{array} $$Step 10 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 1 } \cdot \color{blue}{ \left( -48 \right) } = \color{blue}{ -48 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{1}&6&5&-29&-25&-5&0\\& & 6& 11& -18& -43& \color{blue}{-48} \\ \hline &6&11&-18&-43&\color{blue}{-48}& \end{array} $$Step 11 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ \left( -48 \right) } = \color{orangered}{ -48 } $
$$ \begin{array}{c|rrrrrr}1&6&5&-29&-25&-5&\color{orangered}{ 0 }\\& & 6& 11& -18& -43& \color{orangered}{-48} \\ \hline &\color{blue}{6}&\color{blue}{11}&\color{blue}{-18}&\color{blue}{-43}&\color{blue}{-48}&\color{orangered}{-48} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 6x^{4}+11x^{3}-18x^{2}-43x-48 } $ with a remainder of $ \color{red}{ -48 } $.