Divide using synthetic division $$ ( x^{3}-19x-30 ) \div ( -x-2 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}2&1&0&-19&-30\\& & 2& 4& \color{black}{-30} \\ \hline &\color{blue}{1}&\color{blue}{2}&\color{blue}{-15}&\color{orangered}{-60} \end{array} $$The solution is:
$$ \frac{ x^{3}-19x-30 }{ x-2 } = \color{blue}{x^{2}+2x-15} \color{red}{~-~} \frac{ \color{red}{ 60 } }{ x-2 } $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x -2 = 0 $ ( $ x = \color{blue}{ 2 } $ ) at the left.
$$ \begin{array}{c|rrrr}\color{blue}{2}&1&0&-19&-30\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}2&\color{orangered}{ 1 }&0&-19&-30\\& & & & \\ \hline &\color{orangered}{1}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ 1 } = \color{blue}{ 2 } $.
$$ \begin{array}{c|rrrr}\color{blue}{2}&1&0&-19&-30\\& & \color{blue}{2} & & \\ \hline &\color{blue}{1}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 2 } = \color{orangered}{ 2 } $
$$ \begin{array}{c|rrrr}2&1&\color{orangered}{ 0 }&-19&-30\\& & \color{orangered}{2} & & \\ \hline &1&\color{orangered}{2}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ 2 } = \color{blue}{ 4 } $.
$$ \begin{array}{c|rrrr}\color{blue}{2}&1&0&-19&-30\\& & 2& \color{blue}{4} & \\ \hline &1&\color{blue}{2}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -19 } + \color{orangered}{ 4 } = \color{orangered}{ -15 } $
$$ \begin{array}{c|rrrr}2&1&0&\color{orangered}{ -19 }&-30\\& & 2& \color{orangered}{4} & \\ \hline &1&2&\color{orangered}{-15}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ \left( -15 \right) } = \color{blue}{ -30 } $.
$$ \begin{array}{c|rrrr}\color{blue}{2}&1&0&-19&-30\\& & 2& 4& \color{blue}{-30} \\ \hline &1&2&\color{blue}{-15}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -30 } + \color{orangered}{ \left( -30 \right) } = \color{orangered}{ -60 } $
$$ \begin{array}{c|rrrr}2&1&0&-19&\color{orangered}{ -30 }\\& & 2& 4& \color{orangered}{-30} \\ \hline &\color{blue}{1}&\color{blue}{2}&\color{blue}{-15}&\color{orangered}{-60} \end{array} $$Bottom line represents the quotient $ \color{blue}{ x^{2}+2x-15 } $ with a remainder of $ \color{red}{ -60 } $.