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