Divide using synthetic division $$ ( 6x^{2}-2x-28 ) \div ( 2x+4 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrr}-4&6&-2&-28\\& & -24& \color{black}{104} \\ \hline &\color{blue}{6}&\color{blue}{-26}&\color{orangered}{76} \end{array} $$The solution is:
$$ \frac{ 6x^{2}-2x-28 }{ x+4 } = \color{blue}{6x-26} ~+~ \frac{ \color{red}{ 76 } }{ x+4 } $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x + 4 = 0 $ ( $ x = \color{blue}{ -4 } $ ) at the left.
$$ \begin{array}{c|rrr}\color{blue}{-4}&6&-2&-28\\& & & \\ \hline &&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrr}-4&\color{orangered}{ 6 }&-2&-28\\& & & \\ \hline &\color{orangered}{6}&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -4 } \cdot \color{blue}{ 6 } = \color{blue}{ -24 } $.
$$ \begin{array}{c|rrr}\color{blue}{-4}&6&-2&-28\\& & \color{blue}{-24} & \\ \hline &\color{blue}{6}&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -2 } + \color{orangered}{ \left( -24 \right) } = \color{orangered}{ -26 } $
$$ \begin{array}{c|rrr}-4&6&\color{orangered}{ -2 }&-28\\& & \color{orangered}{-24} & \\ \hline &6&\color{orangered}{-26}& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -4 } \cdot \color{blue}{ \left( -26 \right) } = \color{blue}{ 104 } $.
$$ \begin{array}{c|rrr}\color{blue}{-4}&6&-2&-28\\& & -24& \color{blue}{104} \\ \hline &6&\color{blue}{-26}& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -28 } + \color{orangered}{ 104 } = \color{orangered}{ 76 } $
$$ \begin{array}{c|rrr}-4&6&-2&\color{orangered}{ -28 }\\& & -24& \color{orangered}{104} \\ \hline &\color{blue}{6}&\color{blue}{-26}&\color{orangered}{76} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 6x-26 } $ with a remainder of $ \color{red}{ 76 } $.