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