Divide using synthetic division $$ ( x^{5}-x^{4}-15x^{3}+25x^{2}+14x-24 ) \div ( x-4 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrrr}4&1&-1&-15&25&14&-24\\& & 4& 12& -12& 52& \color{black}{264} \\ \hline &\color{blue}{1}&\color{blue}{3}&\color{blue}{-3}&\color{blue}{13}&\color{blue}{66}&\color{orangered}{240} \end{array} $$The solution is:
$$ \frac{ x^{5}-x^{4}-15x^{3}+25x^{2}+14x-24 }{ x-4 } = \color{blue}{x^{4}+3x^{3}-3x^{2}+13x+66} ~+~ \frac{ \color{red}{ 240 } }{ x-4 } $$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}{ 4 } = \color{orangered}{ 3 } $
$$ \begin{array}{c|rrrrrr}4&1&\color{orangered}{ -1 }&-15&25&14&-24\\& & \color{orangered}{4} & & & & \\ \hline &1&\color{orangered}{3}&&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 4 } \cdot \color{blue}{ 3 } = \color{blue}{ 12 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{4}&1&-1&-15&25&14&-24\\& & 4& \color{blue}{12} & & & \\ \hline &1&\color{blue}{3}&&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -15 } + \color{orangered}{ 12 } = \color{orangered}{ -3 } $
$$ \begin{array}{c|rrrrrr}4&1&-1&\color{orangered}{ -15 }&25&14&-24\\& & 4& \color{orangered}{12} & & & \\ \hline &1&3&\color{orangered}{-3}&&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 4 } \cdot \color{blue}{ \left( -3 \right) } = \color{blue}{ -12 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{4}&1&-1&-15&25&14&-24\\& & 4& 12& \color{blue}{-12} & & \\ \hline &1&3&\color{blue}{-3}&&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 25 } + \color{orangered}{ \left( -12 \right) } = \color{orangered}{ 13 } $
$$ \begin{array}{c|rrrrrr}4&1&-1&-15&\color{orangered}{ 25 }&14&-24\\& & 4& 12& \color{orangered}{-12} & & \\ \hline &1&3&-3&\color{orangered}{13}&& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 4 } \cdot \color{blue}{ 13 } = \color{blue}{ 52 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{4}&1&-1&-15&25&14&-24\\& & 4& 12& -12& \color{blue}{52} & \\ \hline &1&3&-3&\color{blue}{13}&& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ 14 } + \color{orangered}{ 52 } = \color{orangered}{ 66 } $
$$ \begin{array}{c|rrrrrr}4&1&-1&-15&25&\color{orangered}{ 14 }&-24\\& & 4& 12& -12& \color{orangered}{52} & \\ \hline &1&3&-3&13&\color{orangered}{66}& \end{array} $$Step 10 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 4 } \cdot \color{blue}{ 66 } = \color{blue}{ 264 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{4}&1&-1&-15&25&14&-24\\& & 4& 12& -12& 52& \color{blue}{264} \\ \hline &1&3&-3&13&\color{blue}{66}& \end{array} $$Step 11 : Add down last column: $ \color{orangered}{ -24 } + \color{orangered}{ 264 } = \color{orangered}{ 240 } $
$$ \begin{array}{c|rrrrrr}4&1&-1&-15&25&14&\color{orangered}{ -24 }\\& & 4& 12& -12& 52& \color{orangered}{264} \\ \hline &\color{blue}{1}&\color{blue}{3}&\color{blue}{-3}&\color{blue}{13}&\color{blue}{66}&\color{orangered}{240} \end{array} $$Bottom line represents the quotient $ \color{blue}{ x^{4}+3x^{3}-3x^{2}+13x+66 } $ with a remainder of $ \color{red}{ 240 } $.