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