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