Divide using synthetic division $$ ( 8x^{2} ) \div ( 2x+1 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrr}-1&8&0&0\\& & -8& \color{black}{8} \\ \hline &\color{blue}{8}&\color{blue}{-8}&\color{orangered}{8} \end{array} $$The solution is:
$$ \frac{ 8x^{2} }{ x+1 } = \color{blue}{8x-8} ~+~ \frac{ \color{red}{ 8 } }{ x+1 } $$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.
$$ \begin{array}{c|rrr}\color{blue}{-1}&8&0&0\\& & & \\ \hline &&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrr}-1&\color{orangered}{ 8 }&0&0\\& & & \\ \hline &\color{orangered}{8}&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -1 } \cdot \color{blue}{ 8 } = \color{blue}{ -8 } $.
$$ \begin{array}{c|rrr}\color{blue}{-1}&8&0&0\\& & \color{blue}{-8} & \\ \hline &\color{blue}{8}&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ \left( -8 \right) } = \color{orangered}{ -8 } $
$$ \begin{array}{c|rrr}-1&8&\color{orangered}{ 0 }&0\\& & \color{orangered}{-8} & \\ \hline &8&\color{orangered}{-8}& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -1 } \cdot \color{blue}{ \left( -8 \right) } = \color{blue}{ 8 } $.
$$ \begin{array}{c|rrr}\color{blue}{-1}&8&0&0\\& & -8& \color{blue}{8} \\ \hline &8&\color{blue}{-8}& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 8 } = \color{orangered}{ 8 } $
$$ \begin{array}{c|rrr}-1&8&0&\color{orangered}{ 0 }\\& & -8& \color{orangered}{8} \\ \hline &\color{blue}{8}&\color{blue}{-8}&\color{orangered}{8} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 8x-8 } $ with a remainder of $ \color{red}{ 8 } $.