Divide using synthetic division $$ ( x^{5}-x^{4}-15x^{3}+25x^{2}+14x-24 ) \div ( x-3 ) $$
The synthetic division table is:
$$ \begin{array}{c|rrrrrr}3&1&-1&-15&25&14&-24\\& & 3& 6& -27& -6& \color{black}{24} \\ \hline &\color{blue}{1}&\color{blue}{2}&\color{blue}{-9}&\color{blue}{-2}&\color{blue}{8}&\color{orangered}{0} \end{array} $$The solution is:
$$ \dfrac{ x^{5}-x^{4}-15x^{3}+25x^{2}+14x-24 }{ x-3 } = \color{blue}{x^{4}+2x^{3}-9x^{2}-2x+8} $$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}{ 3 } = \color{orangered}{ 2 } $
$$ \begin{array}{c|rrrrrr}3&1&\color{orangered}{ -1 }&-15&25&14&-24\\& & \color{orangered}{3} & & & & \\ \hline &1&\color{orangered}{2}&&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 3 } \cdot \color{blue}{ 2 } = \color{blue}{ 6 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{3}&1&-1&-15&25&14&-24\\& & 3& \color{blue}{6} & & & \\ \hline &1&\color{blue}{2}&&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -15 } + \color{orangered}{ 6 } = \color{orangered}{ -9 } $
$$ \begin{array}{c|rrrrrr}3&1&-1&\color{orangered}{ -15 }&25&14&-24\\& & 3& \color{orangered}{6} & & & \\ \hline &1&2&\color{orangered}{-9}&&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 3 } \cdot \color{blue}{ \left( -9 \right) } = \color{blue}{ -27 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{3}&1&-1&-15&25&14&-24\\& & 3& 6& \color{blue}{-27} & & \\ \hline &1&2&\color{blue}{-9}&&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 25 } + \color{orangered}{ \left( -27 \right) } = \color{orangered}{ -2 } $
$$ \begin{array}{c|rrrrrr}3&1&-1&-15&\color{orangered}{ 25 }&14&-24\\& & 3& 6& \color{orangered}{-27} & & \\ \hline &1&2&-9&\color{orangered}{-2}&& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 3 } \cdot \color{blue}{ \left( -2 \right) } = \color{blue}{ -6 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{3}&1&-1&-15&25&14&-24\\& & 3& 6& -27& \color{blue}{-6} & \\ \hline &1&2&-9&\color{blue}{-2}&& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ 14 } + \color{orangered}{ \left( -6 \right) } = \color{orangered}{ 8 } $
$$ \begin{array}{c|rrrrrr}3&1&-1&-15&25&\color{orangered}{ 14 }&-24\\& & 3& 6& -27& \color{orangered}{-6} & \\ \hline &1&2&-9&-2&\color{orangered}{8}& \end{array} $$Step 10 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 3 } \cdot \color{blue}{ 8 } = \color{blue}{ 24 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{3}&1&-1&-15&25&14&-24\\& & 3& 6& -27& -6& \color{blue}{24} \\ \hline &1&2&-9&-2&\color{blue}{8}& \end{array} $$Step 11 : Add down last column: $ \color{orangered}{ -24 } + \color{orangered}{ 24 } = \color{orangered}{ 0 } $
$$ \begin{array}{c|rrrrrr}3&1&-1&-15&25&14&\color{orangered}{ -24 }\\& & 3& 6& -27& -6& \color{orangered}{24} \\ \hline &\color{blue}{1}&\color{blue}{2}&\color{blue}{-9}&\color{blue}{-2}&\color{blue}{8}&\color{orangered}{0} \end{array} $$Bottom line represents the quotient $ \color{blue}{ x^{4}+2x^{3}-9x^{2}-2x+8 } $ with a remainder of $ \color{red}{ 0 } $.