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