Divide using synthetic division $$ ( x^{3}-10x^{2}+19x+30 ) \div ( x+6 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}-6&1&-10&19&30\\& & -6& 96& \color{black}{-690} \\ \hline &\color{blue}{1}&\color{blue}{-16}&\color{blue}{115}&\color{orangered}{-660} \end{array} $$The solution is:
$$ \frac{ x^{3}-10x^{2}+19x+30 }{ x+6 } = \color{blue}{x^{2}-16x+115} \color{red}{~-~} \frac{ \color{red}{ 660 } }{ x+6 } $$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.
$$ \begin{array}{c|rrrr}\color{blue}{-6}&1&-10&19&30\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}-6&\color{orangered}{ 1 }&-10&19&30\\& & & & \\ \hline &\color{orangered}{1}&&& \end{array} $$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 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-6}&1&-10&19&30\\& & \color{blue}{-6} & & \\ \hline &\color{blue}{1}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -10 } + \color{orangered}{ \left( -6 \right) } = \color{orangered}{ -16 } $
$$ \begin{array}{c|rrrr}-6&1&\color{orangered}{ -10 }&19&30\\& & \color{orangered}{-6} & & \\ \hline &1&\color{orangered}{-16}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -6 } \cdot \color{blue}{ \left( -16 \right) } = \color{blue}{ 96 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-6}&1&-10&19&30\\& & -6& \color{blue}{96} & \\ \hline &1&\color{blue}{-16}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 19 } + \color{orangered}{ 96 } = \color{orangered}{ 115 } $
$$ \begin{array}{c|rrrr}-6&1&-10&\color{orangered}{ 19 }&30\\& & -6& \color{orangered}{96} & \\ \hline &1&-16&\color{orangered}{115}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -6 } \cdot \color{blue}{ 115 } = \color{blue}{ -690 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-6}&1&-10&19&30\\& & -6& 96& \color{blue}{-690} \\ \hline &1&-16&\color{blue}{115}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 30 } + \color{orangered}{ \left( -690 \right) } = \color{orangered}{ -660 } $
$$ \begin{array}{c|rrrr}-6&1&-10&19&\color{orangered}{ 30 }\\& & -6& 96& \color{orangered}{-690} \\ \hline &\color{blue}{1}&\color{blue}{-16}&\color{blue}{115}&\color{orangered}{-660} \end{array} $$Bottom line represents the quotient $ \color{blue}{ x^{2}-16x+115 } $ with a remainder of $ \color{red}{ -660 } $.