Divide using synthetic division $$ ( x^{3}+3x^{2}-28x-54 ) \div ( x+6 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}-6&1&3&-28&-54\\& & -6& 18& \color{black}{60} \\ \hline &\color{blue}{1}&\color{blue}{-3}&\color{blue}{-10}&\color{orangered}{6} \end{array} $$The solution is:
$$ \frac{ x^{3}+3x^{2}-28x-54 }{ x+6 } = \color{blue}{x^{2}-3x-10} ~+~ \frac{ \color{red}{ 6 } }{ x+6 } $$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.
$$ \begin{array}{c|rrrr}\color{blue}{-6}&1&3&-28&-54\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}-6&\color{orangered}{ 1 }&3&-28&-54\\& & & & \\ \hline &\color{orangered}{1}&&& \end{array} $$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 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-6}&1&3&-28&-54\\& & \color{blue}{-6} & & \\ \hline &\color{blue}{1}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 3 } + \color{orangered}{ \left( -6 \right) } = \color{orangered}{ -3 } $
$$ \begin{array}{c|rrrr}-6&1&\color{orangered}{ 3 }&-28&-54\\& & \color{orangered}{-6} & & \\ \hline &1&\color{orangered}{-3}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -6 } \cdot \color{blue}{ \left( -3 \right) } = \color{blue}{ 18 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-6}&1&3&-28&-54\\& & -6& \color{blue}{18} & \\ \hline &1&\color{blue}{-3}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -28 } + \color{orangered}{ 18 } = \color{orangered}{ -10 } $
$$ \begin{array}{c|rrrr}-6&1&3&\color{orangered}{ -28 }&-54\\& & -6& \color{orangered}{18} & \\ \hline &1&-3&\color{orangered}{-10}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -6 } \cdot \color{blue}{ \left( -10 \right) } = \color{blue}{ 60 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-6}&1&3&-28&-54\\& & -6& 18& \color{blue}{60} \\ \hline &1&-3&\color{blue}{-10}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -54 } + \color{orangered}{ 60 } = \color{orangered}{ 6 } $
$$ \begin{array}{c|rrrr}-6&1&3&-28&\color{orangered}{ -54 }\\& & -6& 18& \color{orangered}{60} \\ \hline &\color{blue}{1}&\color{blue}{-3}&\color{blue}{-10}&\color{orangered}{6} \end{array} $$Bottom line represents the quotient $ \color{blue}{ x^{2}-3x-10 } $ with a remainder of $ \color{red}{ 6 } $.