Divide using synthetic division $$ ( x^{3}-2x^{2}-9x+4 ) \div ( x+4 ) $$
The synthetic division table is:
$$ \begin{array}{c|rrrr}-4&1&-2&-9&4\\& & -4& 24& \color{black}{-60} \\ \hline &\color{blue}{1}&\color{blue}{-6}&\color{blue}{15}&\color{orangered}{-56} \end{array} $$The solution is:
$$ \dfrac{ x^{3}-2x^{2}-9x+4 }{ x+4 } = \color{blue}{x^{2}-6x+15} \color{red}{~-~} \dfrac{ \color{red}{ 56 } }{ x+4 } $$Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x + 4 = 0 $ ( $ x = \color{blue}{ -4 } $ ) at the left.
$$ \begin{array}{c|rrrr}\color{blue}{-4}&1&-2&-9&4\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}-4&\color{orangered}{ 1 }&-2&-9&4\\& & & & \\ \hline &\color{orangered}{1}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -4 } \cdot \color{blue}{ 1 } = \color{blue}{ -4 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-4}&1&-2&-9&4\\& & \color{blue}{-4} & & \\ \hline &\color{blue}{1}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -2 } + \color{orangered}{ \left( -4 \right) } = \color{orangered}{ -6 } $
$$ \begin{array}{c|rrrr}-4&1&\color{orangered}{ -2 }&-9&4\\& & \color{orangered}{-4} & & \\ \hline &1&\color{orangered}{-6}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -4 } \cdot \color{blue}{ \left( -6 \right) } = \color{blue}{ 24 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-4}&1&-2&-9&4\\& & -4& \color{blue}{24} & \\ \hline &1&\color{blue}{-6}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -9 } + \color{orangered}{ 24 } = \color{orangered}{ 15 } $
$$ \begin{array}{c|rrrr}-4&1&-2&\color{orangered}{ -9 }&4\\& & -4& \color{orangered}{24} & \\ \hline &1&-6&\color{orangered}{15}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -4 } \cdot \color{blue}{ 15 } = \color{blue}{ -60 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-4}&1&-2&-9&4\\& & -4& 24& \color{blue}{-60} \\ \hline &1&-6&\color{blue}{15}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 4 } + \color{orangered}{ \left( -60 \right) } = \color{orangered}{ -56 } $
$$ \begin{array}{c|rrrr}-4&1&-2&-9&\color{orangered}{ 4 }\\& & -4& 24& \color{orangered}{-60} \\ \hline &\color{blue}{1}&\color{blue}{-6}&\color{blue}{15}&\color{orangered}{-56} \end{array} $$Bottom line represents the quotient $ \color{blue}{ x^{2}-6x+15 } $ with a remainder of $ \color{red}{ -56 } $.