Divide using synthetic division $$ ( x^{3}-45x-66 ) \div ( x+6 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}-6&1&0&-45&-66\\& & -6& 36& \color{black}{54} \\ \hline &\color{blue}{1}&\color{blue}{-6}&\color{blue}{-9}&\color{orangered}{-12} \end{array} $$The solution is:
$$ \frac{ x^{3}-45x-66 }{ x+6 } = \color{blue}{x^{2}-6x-9} \color{red}{~-~} \frac{ \color{red}{ 12 } }{ x+6 } $$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.
$$ \begin{array}{c|rrrr}\color{blue}{-6}&1&0&-45&-66\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}-6&\color{orangered}{ 1 }&0&-45&-66\\& & & & \\ \hline &\color{orangered}{1}&&& \end{array} $$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 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-6}&1&0&-45&-66\\& & \color{blue}{-6} & & \\ \hline &\color{blue}{1}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ \left( -6 \right) } = \color{orangered}{ -6 } $
$$ \begin{array}{c|rrrr}-6&1&\color{orangered}{ 0 }&-45&-66\\& & \color{orangered}{-6} & & \\ \hline &1&\color{orangered}{-6}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -6 } \cdot \color{blue}{ \left( -6 \right) } = \color{blue}{ 36 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-6}&1&0&-45&-66\\& & -6& \color{blue}{36} & \\ \hline &1&\color{blue}{-6}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -45 } + \color{orangered}{ 36 } = \color{orangered}{ -9 } $
$$ \begin{array}{c|rrrr}-6&1&0&\color{orangered}{ -45 }&-66\\& & -6& \color{orangered}{36} & \\ \hline &1&-6&\color{orangered}{-9}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -6 } \cdot \color{blue}{ \left( -9 \right) } = \color{blue}{ 54 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-6}&1&0&-45&-66\\& & -6& 36& \color{blue}{54} \\ \hline &1&-6&\color{blue}{-9}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -66 } + \color{orangered}{ 54 } = \color{orangered}{ -12 } $
$$ \begin{array}{c|rrrr}-6&1&0&-45&\color{orangered}{ -66 }\\& & -6& 36& \color{orangered}{54} \\ \hline &\color{blue}{1}&\color{blue}{-6}&\color{blue}{-9}&\color{orangered}{-12} \end{array} $$Bottom line represents the quotient $ \color{blue}{ x^{2}-6x-9 } $ with a remainder of $ \color{red}{ -12 } $.