Divide using synthetic division $$ ( 20x^{2}-26x-18 ) \div ( 4x+2 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrr}-2&20&-26&-18\\& & -40& \color{black}{132} \\ \hline &\color{blue}{20}&\color{blue}{-66}&\color{orangered}{114} \end{array} $$The solution is:
$$ \frac{ 20x^{2}-26x-18 }{ x+2 } = \color{blue}{20x-66} ~+~ \frac{ \color{red}{ 114 } }{ x+2 } $$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.
$$ \begin{array}{c|rrr}\color{blue}{-2}&20&-26&-18\\& & & \\ \hline &&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrr}-2&\color{orangered}{ 20 }&-26&-18\\& & & \\ \hline &\color{orangered}{20}&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ 20 } = \color{blue}{ -40 } $.
$$ \begin{array}{c|rrr}\color{blue}{-2}&20&-26&-18\\& & \color{blue}{-40} & \\ \hline &\color{blue}{20}&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -26 } + \color{orangered}{ \left( -40 \right) } = \color{orangered}{ -66 } $
$$ \begin{array}{c|rrr}-2&20&\color{orangered}{ -26 }&-18\\& & \color{orangered}{-40} & \\ \hline &20&\color{orangered}{-66}& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ \left( -66 \right) } = \color{blue}{ 132 } $.
$$ \begin{array}{c|rrr}\color{blue}{-2}&20&-26&-18\\& & -40& \color{blue}{132} \\ \hline &20&\color{blue}{-66}& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -18 } + \color{orangered}{ 132 } = \color{orangered}{ 114 } $
$$ \begin{array}{c|rrr}-2&20&-26&\color{orangered}{ -18 }\\& & -40& \color{orangered}{132} \\ \hline &\color{blue}{20}&\color{blue}{-66}&\color{orangered}{114} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 20x-66 } $ with a remainder of $ \color{red}{ 114 } $.