Divide using synthetic division $$ ( 6x^{2}+10x-24 ) \div ( 2x+6 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrr}-6&6&10&-24\\& & -36& \color{black}{156} \\ \hline &\color{blue}{6}&\color{blue}{-26}&\color{orangered}{132} \end{array} $$The solution is:
$$ \frac{ 6x^{2}+10x-24 }{ x+6 } = \color{blue}{6x-26} ~+~ \frac{ \color{red}{ 132 } }{ x+6 } $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x + 6 = 0 $ ( $ x = \color{blue}{ -6 } $ ) at the left.
$$ \begin{array}{c|rrr}\color{blue}{-6}&6&10&-24\\& & & \\ \hline &&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrr}-6&\color{orangered}{ 6 }&10&-24\\& & & \\ \hline &\color{orangered}{6}&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -6 } \cdot \color{blue}{ 6 } = \color{blue}{ -36 } $.
$$ \begin{array}{c|rrr}\color{blue}{-6}&6&10&-24\\& & \color{blue}{-36} & \\ \hline &\color{blue}{6}&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 10 } + \color{orangered}{ \left( -36 \right) } = \color{orangered}{ -26 } $
$$ \begin{array}{c|rrr}-6&6&\color{orangered}{ 10 }&-24\\& & \color{orangered}{-36} & \\ \hline &6&\color{orangered}{-26}& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -6 } \cdot \color{blue}{ \left( -26 \right) } = \color{blue}{ 156 } $.
$$ \begin{array}{c|rrr}\color{blue}{-6}&6&10&-24\\& & -36& \color{blue}{156} \\ \hline &6&\color{blue}{-26}& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -24 } + \color{orangered}{ 156 } = \color{orangered}{ 132 } $
$$ \begin{array}{c|rrr}-6&6&10&\color{orangered}{ -24 }\\& & -36& \color{orangered}{156} \\ \hline &\color{blue}{6}&\color{blue}{-26}&\color{orangered}{132} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 6x-26 } $ with a remainder of $ \color{red}{ 132 } $.