Divide using synthetic division $$ ( x^{3}+12x^{2}+41x+30 ) \div ( x-6 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}6&1&12&41&30\\& & 6& 108& \color{black}{894} \\ \hline &\color{blue}{1}&\color{blue}{18}&\color{blue}{149}&\color{orangered}{924} \end{array} $$The solution is:
$$ \frac{ x^{3}+12x^{2}+41x+30 }{ x-6 } = \color{blue}{x^{2}+18x+149} ~+~ \frac{ \color{red}{ 924 } }{ x-6 } $$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.
$$ \begin{array}{c|rrrr}\color{blue}{6}&1&12&41&30\\& & & & \\ \hline &&&& \end{array} $$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}{ 6 } = \color{orangered}{ 18 } $
$$ \begin{array}{c|rrrr}6&1&\color{orangered}{ 12 }&41&30\\& & \color{orangered}{6} & & \\ \hline &1&\color{orangered}{18}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 6 } \cdot \color{blue}{ 18 } = \color{blue}{ 108 } $.
$$ \begin{array}{c|rrrr}\color{blue}{6}&1&12&41&30\\& & 6& \color{blue}{108} & \\ \hline &1&\color{blue}{18}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 41 } + \color{orangered}{ 108 } = \color{orangered}{ 149 } $
$$ \begin{array}{c|rrrr}6&1&12&\color{orangered}{ 41 }&30\\& & 6& \color{orangered}{108} & \\ \hline &1&18&\color{orangered}{149}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 6 } \cdot \color{blue}{ 149 } = \color{blue}{ 894 } $.
$$ \begin{array}{c|rrrr}\color{blue}{6}&1&12&41&30\\& & 6& 108& \color{blue}{894} \\ \hline &1&18&\color{blue}{149}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 30 } + \color{orangered}{ 894 } = \color{orangered}{ 924 } $
$$ \begin{array}{c|rrrr}6&1&12&41&\color{orangered}{ 30 }\\& & 6& 108& \color{orangered}{894} \\ \hline &\color{blue}{1}&\color{blue}{18}&\color{blue}{149}&\color{orangered}{924} \end{array} $$Bottom line represents the quotient $ \color{blue}{ x^{2}+18x+149 } $ with a remainder of $ \color{red}{ 924 } $.