Divide using synthetic division $$ ( 4x^{4}-9x^{3}+13x^{2}+x-13 ) \div ( x-1 ) $$
The synthetic division table is:
$$ \begin{array}{c|rrrrr}1&4&-9&13&1&-13\\& & 4& -5& 8& \color{black}{9} \\ \hline &\color{blue}{4}&\color{blue}{-5}&\color{blue}{8}&\color{blue}{9}&\color{orangered}{-4} \end{array} $$The solution is:
$$ \dfrac{ 4x^{4}-9x^{3}+13x^{2}+x-13 }{ x-1 } = \color{blue}{4x^{3}-5x^{2}+8x+9} \color{red}{~-~} \dfrac{ \color{red}{ 4 } }{ x-1 } $$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.
$$ \begin{array}{c|rrrrr}\color{blue}{1}&4&-9&13&1&-13\\& & & & & \\ \hline &&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrr}1&\color{orangered}{ 4 }&-9&13&1&-13\\& & & & & \\ \hline &\color{orangered}{4}&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 1 } \cdot \color{blue}{ 4 } = \color{blue}{ 4 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{1}&4&-9&13&1&-13\\& & \color{blue}{4} & & & \\ \hline &\color{blue}{4}&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -9 } + \color{orangered}{ 4 } = \color{orangered}{ -5 } $
$$ \begin{array}{c|rrrrr}1&4&\color{orangered}{ -9 }&13&1&-13\\& & \color{orangered}{4} & & & \\ \hline &4&\color{orangered}{-5}&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 1 } \cdot \color{blue}{ \left( -5 \right) } = \color{blue}{ -5 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{1}&4&-9&13&1&-13\\& & 4& \color{blue}{-5} & & \\ \hline &4&\color{blue}{-5}&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 13 } + \color{orangered}{ \left( -5 \right) } = \color{orangered}{ 8 } $
$$ \begin{array}{c|rrrrr}1&4&-9&\color{orangered}{ 13 }&1&-13\\& & 4& \color{orangered}{-5} & & \\ \hline &4&-5&\color{orangered}{8}&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 1 } \cdot \color{blue}{ 8 } = \color{blue}{ 8 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{1}&4&-9&13&1&-13\\& & 4& -5& \color{blue}{8} & \\ \hline &4&-5&\color{blue}{8}&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 1 } + \color{orangered}{ 8 } = \color{orangered}{ 9 } $
$$ \begin{array}{c|rrrrr}1&4&-9&13&\color{orangered}{ 1 }&-13\\& & 4& -5& \color{orangered}{8} & \\ \hline &4&-5&8&\color{orangered}{9}& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 1 } \cdot \color{blue}{ 9 } = \color{blue}{ 9 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{1}&4&-9&13&1&-13\\& & 4& -5& 8& \color{blue}{9} \\ \hline &4&-5&8&\color{blue}{9}& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ -13 } + \color{orangered}{ 9 } = \color{orangered}{ -4 } $
$$ \begin{array}{c|rrrrr}1&4&-9&13&1&\color{orangered}{ -13 }\\& & 4& -5& 8& \color{orangered}{9} \\ \hline &\color{blue}{4}&\color{blue}{-5}&\color{blue}{8}&\color{blue}{9}&\color{orangered}{-4} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 4x^{3}-5x^{2}+8x+9 } $ with a remainder of $ \color{red}{ -4 } $.