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