Divide using synthetic division $$ ( x^{3}+6x^{2}-x-30 ) \div ( x+3 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}-3&1&6&-1&-30\\& & -3& -9& \color{black}{30} \\ \hline &\color{blue}{1}&\color{blue}{3}&\color{blue}{-10}&\color{orangered}{0} \end{array} $$The solution is:
$$ \frac{ x^{3}+6x^{2}-x-30 }{ x+3 } = \color{blue}{x^{2}+3x-10} $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x + 3 = 0 $ ( $ x = \color{blue}{ -3 } $ ) at the left.
$$ \begin{array}{c|rrrr}\color{blue}{-3}&1&6&-1&-30\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}-3&\color{orangered}{ 1 }&6&-1&-30\\& & & & \\ \hline &\color{orangered}{1}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -3 } \cdot \color{blue}{ 1 } = \color{blue}{ -3 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-3}&1&6&-1&-30\\& & \color{blue}{-3} & & \\ \hline &\color{blue}{1}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 6 } + \color{orangered}{ \left( -3 \right) } = \color{orangered}{ 3 } $
$$ \begin{array}{c|rrrr}-3&1&\color{orangered}{ 6 }&-1&-30\\& & \color{orangered}{-3} & & \\ \hline &1&\color{orangered}{3}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -3 } \cdot \color{blue}{ 3 } = \color{blue}{ -9 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-3}&1&6&-1&-30\\& & -3& \color{blue}{-9} & \\ \hline &1&\color{blue}{3}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -1 } + \color{orangered}{ \left( -9 \right) } = \color{orangered}{ -10 } $
$$ \begin{array}{c|rrrr}-3&1&6&\color{orangered}{ -1 }&-30\\& & -3& \color{orangered}{-9} & \\ \hline &1&3&\color{orangered}{-10}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -3 } \cdot \color{blue}{ \left( -10 \right) } = \color{blue}{ 30 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-3}&1&6&-1&-30\\& & -3& -9& \color{blue}{30} \\ \hline &1&3&\color{blue}{-10}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -30 } + \color{orangered}{ 30 } = \color{orangered}{ 0 } $
$$ \begin{array}{c|rrrr}-3&1&6&-1&\color{orangered}{ -30 }\\& & -3& -9& \color{orangered}{30} \\ \hline &\color{blue}{1}&\color{blue}{3}&\color{blue}{-10}&\color{orangered}{0} \end{array} $$Bottom line represents the quotient $ \color{blue}{ x^{2}+3x-10 } $ with a remainder of $ \color{red}{ 0 } $.