Divide using synthetic division $$ ( -x^{3}+19x+30 ) \div ( x+2 ) $$
The synthetic division table is:
$$ \begin{array}{c|rrrr}-2&-1&0&19&30\\& & 2& -4& \color{black}{-30} \\ \hline &\color{blue}{-1}&\color{blue}{2}&\color{blue}{15}&\color{orangered}{0} \end{array} $$The solution is:
$$ \dfrac{ -x^{3}+19x+30 }{ x+2 } = \color{blue}{-x^{2}+2x+15} $$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}{ \left( -1 \right) } = \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}{ \left( -4 \right) } = \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}{ 15 } = \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}{ 0 } $
$$ \begin{array}{c|rrrr}-2&-1&0&19&\color{orangered}{ 30 }\\& & 2& -4& \color{orangered}{-30} \\ \hline &\color{blue}{-1}&\color{blue}{2}&\color{blue}{15}&\color{orangered}{0} \end{array} $$Bottom line represents the quotient $ \color{blue}{ -x^{2}+2x+15 } $ with a remainder of $ \color{red}{ 0 } $.