Divide using synthetic division $$ ( -20x^{2}-13x+2 ) \div ( 5x-2 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrr}2&-20&-13&2\\& & -40& \color{black}{-106} \\ \hline &\color{blue}{-20}&\color{blue}{-53}&\color{orangered}{-104} \end{array} $$The solution is:
$$ \frac{ -20x^{2}-13x+2 }{ x-2 } = \color{blue}{-20x-53} \color{red}{~-~} \frac{ \color{red}{ 104 } }{ x-2 } $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x -2 = 0 $ ( $ x = \color{blue}{ 2 } $ ) at the left.
$$ \begin{array}{c|rrr}\color{blue}{2}&-20&-13&2\\& & & \\ \hline &&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrr}2&\color{orangered}{ -20 }&-13&2\\& & & \\ \hline &\color{orangered}{-20}&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ \left( -20 \right) } = \color{blue}{ -40 } $.
$$ \begin{array}{c|rrr}\color{blue}{2}&-20&-13&2\\& & \color{blue}{-40} & \\ \hline &\color{blue}{-20}&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -13 } + \color{orangered}{ \left( -40 \right) } = \color{orangered}{ -53 } $
$$ \begin{array}{c|rrr}2&-20&\color{orangered}{ -13 }&2\\& & \color{orangered}{-40} & \\ \hline &-20&\color{orangered}{-53}& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ \left( -53 \right) } = \color{blue}{ -106 } $.
$$ \begin{array}{c|rrr}\color{blue}{2}&-20&-13&2\\& & -40& \color{blue}{-106} \\ \hline &-20&\color{blue}{-53}& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 2 } + \color{orangered}{ \left( -106 \right) } = \color{orangered}{ -104 } $
$$ \begin{array}{c|rrr}2&-20&-13&\color{orangered}{ 2 }\\& & -40& \color{orangered}{-106} \\ \hline &\color{blue}{-20}&\color{blue}{-53}&\color{orangered}{-104} \end{array} $$Bottom line represents the quotient $ \color{blue}{ -20x-53 } $ with a remainder of $ \color{red}{ -104 } $.