Divide using synthetic division $$ ( x^{3}+12x^{2}+41x+30 ) \div ( x+6 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}-6&1&12&41&30\\& & -6& -36& \color{black}{-30} \\ \hline &\color{blue}{1}&\color{blue}{6}&\color{blue}{5}&\color{orangered}{0} \end{array} $$The solution is:
$$ \frac{ x^{3}+12x^{2}+41x+30 }{ x+6 } = \color{blue}{x^{2}+6x+5} $$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.
$$ \begin{array}{c|rrrr}\color{blue}{-6}&1&12&41&30\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}-6&\color{orangered}{ 1 }&12&41&30\\& & & & \\ \hline &\color{orangered}{1}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -6 } \cdot \color{blue}{ 1 } = \color{blue}{ -6 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-6}&1&12&41&30\\& & \color{blue}{-6} & & \\ \hline &\color{blue}{1}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 12 } + \color{orangered}{ \left( -6 \right) } = \color{orangered}{ 6 } $
$$ \begin{array}{c|rrrr}-6&1&\color{orangered}{ 12 }&41&30\\& & \color{orangered}{-6} & & \\ \hline &1&\color{orangered}{6}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -6 } \cdot \color{blue}{ 6 } = \color{blue}{ -36 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-6}&1&12&41&30\\& & -6& \color{blue}{-36} & \\ \hline &1&\color{blue}{6}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 41 } + \color{orangered}{ \left( -36 \right) } = \color{orangered}{ 5 } $
$$ \begin{array}{c|rrrr}-6&1&12&\color{orangered}{ 41 }&30\\& & -6& \color{orangered}{-36} & \\ \hline &1&6&\color{orangered}{5}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -6 } \cdot \color{blue}{ 5 } = \color{blue}{ -30 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-6}&1&12&41&30\\& & -6& -36& \color{blue}{-30} \\ \hline &1&6&\color{blue}{5}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 30 } + \color{orangered}{ \left( -30 \right) } = \color{orangered}{ 0 } $
$$ \begin{array}{c|rrrr}-6&1&12&41&\color{orangered}{ 30 }\\& & -6& -36& \color{orangered}{-30} \\ \hline &\color{blue}{1}&\color{blue}{6}&\color{blue}{5}&\color{orangered}{0} \end{array} $$Bottom line represents the quotient $ \color{blue}{ x^{2}+6x+5 } $ with a remainder of $ \color{red}{ 0 } $.