Divide using synthetic division $$ ( -5x^{2}+10x-1 ) \div ( 3x-1 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrr}1&-5&10&-1\\& & -5& \color{black}{5} \\ \hline &\color{blue}{-5}&\color{blue}{5}&\color{orangered}{4} \end{array} $$The solution is:
$$ \frac{ -5x^{2}+10x-1 }{ x-1 } = \color{blue}{-5x+5} ~+~ \frac{ \color{red}{ 4 } }{ x-1 } $$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.
$$ \begin{array}{c|rrr}\color{blue}{1}&-5&10&-1\\& & & \\ \hline &&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrr}1&\color{orangered}{ -5 }&10&-1\\& & & \\ \hline &\color{orangered}{-5}&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 1 } \cdot \color{blue}{ \left( -5 \right) } = \color{blue}{ -5 } $.
$$ \begin{array}{c|rrr}\color{blue}{1}&-5&10&-1\\& & \color{blue}{-5} & \\ \hline &\color{blue}{-5}&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 10 } + \color{orangered}{ \left( -5 \right) } = \color{orangered}{ 5 } $
$$ \begin{array}{c|rrr}1&-5&\color{orangered}{ 10 }&-1\\& & \color{orangered}{-5} & \\ \hline &-5&\color{orangered}{5}& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 1 } \cdot \color{blue}{ 5 } = \color{blue}{ 5 } $.
$$ \begin{array}{c|rrr}\color{blue}{1}&-5&10&-1\\& & -5& \color{blue}{5} \\ \hline &-5&\color{blue}{5}& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -1 } + \color{orangered}{ 5 } = \color{orangered}{ 4 } $
$$ \begin{array}{c|rrr}1&-5&10&\color{orangered}{ -1 }\\& & -5& \color{orangered}{5} \\ \hline &\color{blue}{-5}&\color{blue}{5}&\color{orangered}{4} \end{array} $$Bottom line represents the quotient $ \color{blue}{ -5x+5 } $ with a remainder of $ \color{red}{ 4 } $.