Divide using synthetic division $$ ( 2x^{2}+6x-8 ) \div ( 2x+8 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrr}-8&2&6&-8\\& & -16& \color{black}{80} \\ \hline &\color{blue}{2}&\color{blue}{-10}&\color{orangered}{72} \end{array} $$The solution is:
$$ \frac{ 2x^{2}+6x-8 }{ x+8 } = \color{blue}{2x-10} ~+~ \frac{ \color{red}{ 72 } }{ x+8 } $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x + 8 = 0 $ ( $ x = \color{blue}{ -8 } $ ) at the left.
$$ \begin{array}{c|rrr}\color{blue}{-8}&2&6&-8\\& & & \\ \hline &&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrr}-8&\color{orangered}{ 2 }&6&-8\\& & & \\ \hline &\color{orangered}{2}&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -8 } \cdot \color{blue}{ 2 } = \color{blue}{ -16 } $.
$$ \begin{array}{c|rrr}\color{blue}{-8}&2&6&-8\\& & \color{blue}{-16} & \\ \hline &\color{blue}{2}&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 6 } + \color{orangered}{ \left( -16 \right) } = \color{orangered}{ -10 } $
$$ \begin{array}{c|rrr}-8&2&\color{orangered}{ 6 }&-8\\& & \color{orangered}{-16} & \\ \hline &2&\color{orangered}{-10}& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -8 } \cdot \color{blue}{ \left( -10 \right) } = \color{blue}{ 80 } $.
$$ \begin{array}{c|rrr}\color{blue}{-8}&2&6&-8\\& & -16& \color{blue}{80} \\ \hline &2&\color{blue}{-10}& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -8 } + \color{orangered}{ 80 } = \color{orangered}{ 72 } $
$$ \begin{array}{c|rrr}-8&2&6&\color{orangered}{ -8 }\\& & -16& \color{orangered}{80} \\ \hline &\color{blue}{2}&\color{blue}{-10}&\color{orangered}{72} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 2x-10 } $ with a remainder of $ \color{red}{ 72 } $.