Divide using synthetic division $$ ( x^{5}-x^{4}-15x^{3}+25x^{2}+14x-24 ) \div ( x+3 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrrr}-3&1&-1&-15&25&14&-24\\& & -3& 12& 9& -102& \color{black}{264} \\ \hline &\color{blue}{1}&\color{blue}{-4}&\color{blue}{-3}&\color{blue}{34}&\color{blue}{-88}&\color{orangered}{240} \end{array} $$The solution is:
$$ \frac{ x^{5}-x^{4}-15x^{3}+25x^{2}+14x-24 }{ x+3 } = \color{blue}{x^{4}-4x^{3}-3x^{2}+34x-88} ~+~ \frac{ \color{red}{ 240 } }{ x+3 } $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x + 3 = 0 $ ( $ x = \color{blue}{ -3 } $ ) at the left.
$$ \begin{array}{c|rrrrrr}\color{blue}{-3}&1&-1&-15&25&14&-24\\& & & & & & \\ \hline &&&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrrr}-3&\color{orangered}{ 1 }&-1&-15&25&14&-24\\& & & & & & \\ \hline &\color{orangered}{1}&&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -3 } \cdot \color{blue}{ 1 } = \color{blue}{ -3 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{-3}&1&-1&-15&25&14&-24\\& & \color{blue}{-3} & & & & \\ \hline &\color{blue}{1}&&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -1 } + \color{orangered}{ \left( -3 \right) } = \color{orangered}{ -4 } $
$$ \begin{array}{c|rrrrrr}-3&1&\color{orangered}{ -1 }&-15&25&14&-24\\& & \color{orangered}{-3} & & & & \\ \hline &1&\color{orangered}{-4}&&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -3 } \cdot \color{blue}{ \left( -4 \right) } = \color{blue}{ 12 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{-3}&1&-1&-15&25&14&-24\\& & -3& \color{blue}{12} & & & \\ \hline &1&\color{blue}{-4}&&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -15 } + \color{orangered}{ 12 } = \color{orangered}{ -3 } $
$$ \begin{array}{c|rrrrrr}-3&1&-1&\color{orangered}{ -15 }&25&14&-24\\& & -3& \color{orangered}{12} & & & \\ \hline &1&-4&\color{orangered}{-3}&&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -3 } \cdot \color{blue}{ \left( -3 \right) } = \color{blue}{ 9 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{-3}&1&-1&-15&25&14&-24\\& & -3& 12& \color{blue}{9} & & \\ \hline &1&-4&\color{blue}{-3}&&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 25 } + \color{orangered}{ 9 } = \color{orangered}{ 34 } $
$$ \begin{array}{c|rrrrrr}-3&1&-1&-15&\color{orangered}{ 25 }&14&-24\\& & -3& 12& \color{orangered}{9} & & \\ \hline &1&-4&-3&\color{orangered}{34}&& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -3 } \cdot \color{blue}{ 34 } = \color{blue}{ -102 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{-3}&1&-1&-15&25&14&-24\\& & -3& 12& 9& \color{blue}{-102} & \\ \hline &1&-4&-3&\color{blue}{34}&& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ 14 } + \color{orangered}{ \left( -102 \right) } = \color{orangered}{ -88 } $
$$ \begin{array}{c|rrrrrr}-3&1&-1&-15&25&\color{orangered}{ 14 }&-24\\& & -3& 12& 9& \color{orangered}{-102} & \\ \hline &1&-4&-3&34&\color{orangered}{-88}& \end{array} $$Step 10 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -3 } \cdot \color{blue}{ \left( -88 \right) } = \color{blue}{ 264 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{-3}&1&-1&-15&25&14&-24\\& & -3& 12& 9& -102& \color{blue}{264} \\ \hline &1&-4&-3&34&\color{blue}{-88}& \end{array} $$Step 11 : Add down last column: $ \color{orangered}{ -24 } + \color{orangered}{ 264 } = \color{orangered}{ 240 } $
$$ \begin{array}{c|rrrrrr}-3&1&-1&-15&25&14&\color{orangered}{ -24 }\\& & -3& 12& 9& -102& \color{orangered}{264} \\ \hline &\color{blue}{1}&\color{blue}{-4}&\color{blue}{-3}&\color{blue}{34}&\color{blue}{-88}&\color{orangered}{240} \end{array} $$Bottom line represents the quotient $ \color{blue}{ x^{4}-4x^{3}-3x^{2}+34x-88 } $ with a remainder of $ \color{red}{ 240 } $.