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