Divide using synthetic division $$ ( 4x^{2}+2x ) \div ( x+7 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrr}-7&4&2&0\\& & -28& \color{black}{182} \\ \hline &\color{blue}{4}&\color{blue}{-26}&\color{orangered}{182} \end{array} $$The solution is:
$$ \frac{ 4x^{2}+2x }{ x+7 } = \color{blue}{4x-26} ~+~ \frac{ \color{red}{ 182 } }{ x+7 } $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x + 7 = 0 $ ( $ x = \color{blue}{ -7 } $ ) at the left.
$$ \begin{array}{c|rrr}\color{blue}{-7}&4&2&0\\& & & \\ \hline &&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrr}-7&\color{orangered}{ 4 }&2&0\\& & & \\ \hline &\color{orangered}{4}&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -7 } \cdot \color{blue}{ 4 } = \color{blue}{ -28 } $.
$$ \begin{array}{c|rrr}\color{blue}{-7}&4&2&0\\& & \color{blue}{-28} & \\ \hline &\color{blue}{4}&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 2 } + \color{orangered}{ \left( -28 \right) } = \color{orangered}{ -26 } $
$$ \begin{array}{c|rrr}-7&4&\color{orangered}{ 2 }&0\\& & \color{orangered}{-28} & \\ \hline &4&\color{orangered}{-26}& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -7 } \cdot \color{blue}{ \left( -26 \right) } = \color{blue}{ 182 } $.
$$ \begin{array}{c|rrr}\color{blue}{-7}&4&2&0\\& & -28& \color{blue}{182} \\ \hline &4&\color{blue}{-26}& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 182 } = \color{orangered}{ 182 } $
$$ \begin{array}{c|rrr}-7&4&2&\color{orangered}{ 0 }\\& & -28& \color{orangered}{182} \\ \hline &\color{blue}{4}&\color{blue}{-26}&\color{orangered}{182} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 4x-26 } $ with a remainder of $ \color{red}{ 182 } $.