Divide using synthetic division $$ ( x^{5}-x^{4}+3x^{3}-3x^{2}+9x-14 ) \div ( x-1 ) $$
The synthetic division table is:
$$ \begin{array}{c|rrrrrr}1&1&-1&3&-3&9&-14\\& & 1& 0& 3& 0& \color{black}{9} \\ \hline &\color{blue}{1}&\color{blue}{0}&\color{blue}{3}&\color{blue}{0}&\color{blue}{9}&\color{orangered}{-5} \end{array} $$The solution is:
$$ \dfrac{ x^{5}-x^{4}+3x^{3}-3x^{2}+9x-14 }{ x-1 } = \color{blue}{x^{4}+3x^{2}+9} \color{red}{~-~} \dfrac{ \color{red}{ 5 } }{ x-1 } $$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.
$$ \begin{array}{c|rrrrrr}\color{blue}{1}&1&-1&3&-3&9&-14\\& & & & & & \\ \hline &&&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrrr}1&\color{orangered}{ 1 }&-1&3&-3&9&-14\\& & & & & & \\ \hline &\color{orangered}{1}&&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 1 } \cdot \color{blue}{ 1 } = \color{blue}{ 1 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{1}&1&-1&3&-3&9&-14\\& & \color{blue}{1} & & & & \\ \hline &\color{blue}{1}&&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -1 } + \color{orangered}{ 1 } = \color{orangered}{ 0 } $
$$ \begin{array}{c|rrrrrr}1&1&\color{orangered}{ -1 }&3&-3&9&-14\\& & \color{orangered}{1} & & & & \\ \hline &1&\color{orangered}{0}&&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 1 } \cdot \color{blue}{ 0 } = \color{blue}{ 0 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{1}&1&-1&3&-3&9&-14\\& & 1& \color{blue}{0} & & & \\ \hline &1&\color{blue}{0}&&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 3 } + \color{orangered}{ 0 } = \color{orangered}{ 3 } $
$$ \begin{array}{c|rrrrrr}1&1&-1&\color{orangered}{ 3 }&-3&9&-14\\& & 1& \color{orangered}{0} & & & \\ \hline &1&0&\color{orangered}{3}&&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 1 } \cdot \color{blue}{ 3 } = \color{blue}{ 3 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{1}&1&-1&3&-3&9&-14\\& & 1& 0& \color{blue}{3} & & \\ \hline &1&0&\color{blue}{3}&&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -3 } + \color{orangered}{ 3 } = \color{orangered}{ 0 } $
$$ \begin{array}{c|rrrrrr}1&1&-1&3&\color{orangered}{ -3 }&9&-14\\& & 1& 0& \color{orangered}{3} & & \\ \hline &1&0&3&\color{orangered}{0}&& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 1 } \cdot \color{blue}{ 0 } = \color{blue}{ 0 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{1}&1&-1&3&-3&9&-14\\& & 1& 0& 3& \color{blue}{0} & \\ \hline &1&0&3&\color{blue}{0}&& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ 9 } + \color{orangered}{ 0 } = \color{orangered}{ 9 } $
$$ \begin{array}{c|rrrrrr}1&1&-1&3&-3&\color{orangered}{ 9 }&-14\\& & 1& 0& 3& \color{orangered}{0} & \\ \hline &1&0&3&0&\color{orangered}{9}& \end{array} $$Step 10 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 1 } \cdot \color{blue}{ 9 } = \color{blue}{ 9 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{1}&1&-1&3&-3&9&-14\\& & 1& 0& 3& 0& \color{blue}{9} \\ \hline &1&0&3&0&\color{blue}{9}& \end{array} $$Step 11 : Add down last column: $ \color{orangered}{ -14 } + \color{orangered}{ 9 } = \color{orangered}{ -5 } $
$$ \begin{array}{c|rrrrrr}1&1&-1&3&-3&9&\color{orangered}{ -14 }\\& & 1& 0& 3& 0& \color{orangered}{9} \\ \hline &\color{blue}{1}&\color{blue}{0}&\color{blue}{3}&\color{blue}{0}&\color{blue}{9}&\color{orangered}{-5} \end{array} $$Bottom line represents the quotient $ \color{blue}{ x^{4}+3x^{2}+9 } $ with a remainder of $ \color{red}{ -5 } $.