Divide using synthetic division $$ ( -14x^{2} ) \div ( 4x+2 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrr}-2&-14&0&0\\& & 28& \color{black}{-56} \\ \hline &\color{blue}{-14}&\color{blue}{28}&\color{orangered}{-56} \end{array} $$The solution is:
$$ \frac{ -14x^{2} }{ x+2 } = \color{blue}{-14x+28} \color{red}{~-~} \frac{ \color{red}{ 56 } }{ x+2 } $$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.
$$ \begin{array}{c|rrr}\color{blue}{-2}&-14&0&0\\& & & \\ \hline &&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrr}-2&\color{orangered}{ -14 }&0&0\\& & & \\ \hline &\color{orangered}{-14}&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ \left( -14 \right) } = \color{blue}{ 28 } $.
$$ \begin{array}{c|rrr}\color{blue}{-2}&-14&0&0\\& & \color{blue}{28} & \\ \hline &\color{blue}{-14}&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 28 } = \color{orangered}{ 28 } $
$$ \begin{array}{c|rrr}-2&-14&\color{orangered}{ 0 }&0\\& & \color{orangered}{28} & \\ \hline &-14&\color{orangered}{28}& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ 28 } = \color{blue}{ -56 } $.
$$ \begin{array}{c|rrr}\color{blue}{-2}&-14&0&0\\& & 28& \color{blue}{-56} \\ \hline &-14&\color{blue}{28}& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ \left( -56 \right) } = \color{orangered}{ -56 } $
$$ \begin{array}{c|rrr}-2&-14&0&\color{orangered}{ 0 }\\& & 28& \color{orangered}{-56} \\ \hline &\color{blue}{-14}&\color{blue}{28}&\color{orangered}{-56} \end{array} $$Bottom line represents the quotient $ \color{blue}{ -14x+28 } $ with a remainder of $ \color{red}{ -56 } $.