Divide using synthetic division $$ ( 6x^{3}+30x^{2}+20x-23 ) \div ( x+4 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}-4&6&30&20&-23\\& & -24& -24& \color{black}{16} \\ \hline &\color{blue}{6}&\color{blue}{6}&\color{blue}{-4}&\color{orangered}{-7} \end{array} $$The solution is:
$$ \frac{ 6x^{3}+30x^{2}+20x-23 }{ x+4 } = \color{blue}{6x^{2}+6x-4} \color{red}{~-~} \frac{ \color{red}{ 7 } }{ x+4 } $$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.
$$ \begin{array}{c|rrrr}\color{blue}{-4}&6&30&20&-23\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}-4&\color{orangered}{ 6 }&30&20&-23\\& & & & \\ \hline &\color{orangered}{6}&&& \end{array} $$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 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-4}&6&30&20&-23\\& & \color{blue}{-24} & & \\ \hline &\color{blue}{6}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 30 } + \color{orangered}{ \left( -24 \right) } = \color{orangered}{ 6 } $
$$ \begin{array}{c|rrrr}-4&6&\color{orangered}{ 30 }&20&-23\\& & \color{orangered}{-24} & & \\ \hline &6&\color{orangered}{6}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -4 } \cdot \color{blue}{ 6 } = \color{blue}{ -24 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-4}&6&30&20&-23\\& & -24& \color{blue}{-24} & \\ \hline &6&\color{blue}{6}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 20 } + \color{orangered}{ \left( -24 \right) } = \color{orangered}{ -4 } $
$$ \begin{array}{c|rrrr}-4&6&30&\color{orangered}{ 20 }&-23\\& & -24& \color{orangered}{-24} & \\ \hline &6&6&\color{orangered}{-4}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -4 } \cdot \color{blue}{ \left( -4 \right) } = \color{blue}{ 16 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-4}&6&30&20&-23\\& & -24& -24& \color{blue}{16} \\ \hline &6&6&\color{blue}{-4}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -23 } + \color{orangered}{ 16 } = \color{orangered}{ -7 } $
$$ \begin{array}{c|rrrr}-4&6&30&20&\color{orangered}{ -23 }\\& & -24& -24& \color{orangered}{16} \\ \hline &\color{blue}{6}&\color{blue}{6}&\color{blue}{-4}&\color{orangered}{-7} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 6x^{2}+6x-4 } $ with a remainder of $ \color{red}{ -7 } $.