Divide using synthetic division $$ ( x^{5}-x^{4}-15x^{3}+25x^{2}+14x-24 ) \div ( x+4 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrrr}-4&1&-1&-15&25&14&-24\\& & -4& 20& -20& -20& \color{black}{24} \\ \hline &\color{blue}{1}&\color{blue}{-5}&\color{blue}{5}&\color{blue}{5}&\color{blue}{-6}&\color{orangered}{0} \end{array} $$The solution is:
$$ \frac{ x^{5}-x^{4}-15x^{3}+25x^{2}+14x-24 }{ x+4 } = \color{blue}{x^{4}-5x^{3}+5x^{2}+5x-6} $$Explanation
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.
$$ \begin{array}{c|rrrrrr}\color{blue}{-4}&1&-1&-15&25&14&-24\\& & & & & & \\ \hline &&&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrrr}-4&\color{orangered}{ 1 }&-1&-15&25&14&-24\\& & & & & & \\ \hline &\color{orangered}{1}&&&&& \end{array} $$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 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{-4}&1&-1&-15&25&14&-24\\& & \color{blue}{-4} & & & & \\ \hline &\color{blue}{1}&&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -1 } + \color{orangered}{ \left( -4 \right) } = \color{orangered}{ -5 } $
$$ \begin{array}{c|rrrrrr}-4&1&\color{orangered}{ -1 }&-15&25&14&-24\\& & \color{orangered}{-4} & & & & \\ \hline &1&\color{orangered}{-5}&&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -4 } \cdot \color{blue}{ \left( -5 \right) } = \color{blue}{ 20 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{-4}&1&-1&-15&25&14&-24\\& & -4& \color{blue}{20} & & & \\ \hline &1&\color{blue}{-5}&&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -15 } + \color{orangered}{ 20 } = \color{orangered}{ 5 } $
$$ \begin{array}{c|rrrrrr}-4&1&-1&\color{orangered}{ -15 }&25&14&-24\\& & -4& \color{orangered}{20} & & & \\ \hline &1&-5&\color{orangered}{5}&&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -4 } \cdot \color{blue}{ 5 } = \color{blue}{ -20 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{-4}&1&-1&-15&25&14&-24\\& & -4& 20& \color{blue}{-20} & & \\ \hline &1&-5&\color{blue}{5}&&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 25 } + \color{orangered}{ \left( -20 \right) } = \color{orangered}{ 5 } $
$$ \begin{array}{c|rrrrrr}-4&1&-1&-15&\color{orangered}{ 25 }&14&-24\\& & -4& 20& \color{orangered}{-20} & & \\ \hline &1&-5&5&\color{orangered}{5}&& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -4 } \cdot \color{blue}{ 5 } = \color{blue}{ -20 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{-4}&1&-1&-15&25&14&-24\\& & -4& 20& -20& \color{blue}{-20} & \\ \hline &1&-5&5&\color{blue}{5}&& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ 14 } + \color{orangered}{ \left( -20 \right) } = \color{orangered}{ -6 } $
$$ \begin{array}{c|rrrrrr}-4&1&-1&-15&25&\color{orangered}{ 14 }&-24\\& & -4& 20& -20& \color{orangered}{-20} & \\ \hline &1&-5&5&5&\color{orangered}{-6}& \end{array} $$Step 10 : 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 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{-4}&1&-1&-15&25&14&-24\\& & -4& 20& -20& -20& \color{blue}{24} \\ \hline &1&-5&5&5&\color{blue}{-6}& \end{array} $$Step 11 : Add down last column: $ \color{orangered}{ -24 } + \color{orangered}{ 24 } = \color{orangered}{ 0 } $
$$ \begin{array}{c|rrrrrr}-4&1&-1&-15&25&14&\color{orangered}{ -24 }\\& & -4& 20& -20& -20& \color{orangered}{24} \\ \hline &\color{blue}{1}&\color{blue}{-5}&\color{blue}{5}&\color{blue}{5}&\color{blue}{-6}&\color{orangered}{0} \end{array} $$Bottom line represents the quotient $ \color{blue}{ x^{4}-5x^{3}+5x^{2}+5x-6 } $ with a remainder of $ \color{red}{ 0 } $.