Divide using synthetic division $$ ( x^{4}+5x^{3}+2x^{2}-20x-24 ) \div ( x+3 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrr}-3&1&5&2&-20&-24\\& & -3& -6& 12& \color{black}{24} \\ \hline &\color{blue}{1}&\color{blue}{2}&\color{blue}{-4}&\color{blue}{-8}&\color{orangered}{0} \end{array} $$The solution is:
$$ \frac{ x^{4}+5x^{3}+2x^{2}-20x-24 }{ x+3 } = \color{blue}{x^{3}+2x^{2}-4x-8} $$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.
$$ \begin{array}{c|rrrrr}\color{blue}{-3}&1&5&2&-20&-24\\& & & & & \\ \hline &&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrr}-3&\color{orangered}{ 1 }&5&2&-20&-24\\& & & & & \\ \hline &\color{orangered}{1}&&&& \end{array} $$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 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-3}&1&5&2&-20&-24\\& & \color{blue}{-3} & & & \\ \hline &\color{blue}{1}&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 5 } + \color{orangered}{ \left( -3 \right) } = \color{orangered}{ 2 } $
$$ \begin{array}{c|rrrrr}-3&1&\color{orangered}{ 5 }&2&-20&-24\\& & \color{orangered}{-3} & & & \\ \hline &1&\color{orangered}{2}&&& \end{array} $$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 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-3}&1&5&2&-20&-24\\& & -3& \color{blue}{-6} & & \\ \hline &1&\color{blue}{2}&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 2 } + \color{orangered}{ \left( -6 \right) } = \color{orangered}{ -4 } $
$$ \begin{array}{c|rrrrr}-3&1&5&\color{orangered}{ 2 }&-20&-24\\& & -3& \color{orangered}{-6} & & \\ \hline &1&2&\color{orangered}{-4}&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -3 } \cdot \color{blue}{ \left( -4 \right) } = \color{blue}{ 12 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-3}&1&5&2&-20&-24\\& & -3& -6& \color{blue}{12} & \\ \hline &1&2&\color{blue}{-4}&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -20 } + \color{orangered}{ 12 } = \color{orangered}{ -8 } $
$$ \begin{array}{c|rrrrr}-3&1&5&2&\color{orangered}{ -20 }&-24\\& & -3& -6& \color{orangered}{12} & \\ \hline &1&2&-4&\color{orangered}{-8}& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -3 } \cdot \color{blue}{ \left( -8 \right) } = \color{blue}{ 24 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-3}&1&5&2&-20&-24\\& & -3& -6& 12& \color{blue}{24} \\ \hline &1&2&-4&\color{blue}{-8}& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ -24 } + \color{orangered}{ 24 } = \color{orangered}{ 0 } $
$$ \begin{array}{c|rrrrr}-3&1&5&2&-20&\color{orangered}{ -24 }\\& & -3& -6& 12& \color{orangered}{24} \\ \hline &\color{blue}{1}&\color{blue}{2}&\color{blue}{-4}&\color{blue}{-8}&\color{orangered}{0} \end{array} $$Bottom line represents the quotient $ \color{blue}{ x^{3}+2x^{2}-4x-8 } $ with a remainder of $ \color{red}{ 0 } $.