Divide using synthetic division $$ ( x^{4}-8x^{3}+12x^{2}+12x-38 ) \div ( x+4 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrr}-4&1&-8&12&12&-38\\& & -4& 48& -240& \color{black}{912} \\ \hline &\color{blue}{1}&\color{blue}{-12}&\color{blue}{60}&\color{blue}{-228}&\color{orangered}{874} \end{array} $$The solution is:
$$ \frac{ x^{4}-8x^{3}+12x^{2}+12x-38 }{ x+4 } = \color{blue}{x^{3}-12x^{2}+60x-228} ~+~ \frac{ \color{red}{ 874 } }{ x+4 } $$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.
$$ \begin{array}{c|rrrrr}\color{blue}{-4}&1&-8&12&12&-38\\& & & & & \\ \hline &&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrr}-4&\color{orangered}{ 1 }&-8&12&12&-38\\& & & & & \\ \hline &\color{orangered}{1}&&&& \end{array} $$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 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-4}&1&-8&12&12&-38\\& & \color{blue}{-4} & & & \\ \hline &\color{blue}{1}&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -8 } + \color{orangered}{ \left( -4 \right) } = \color{orangered}{ -12 } $
$$ \begin{array}{c|rrrrr}-4&1&\color{orangered}{ -8 }&12&12&-38\\& & \color{orangered}{-4} & & & \\ \hline &1&\color{orangered}{-12}&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -4 } \cdot \color{blue}{ \left( -12 \right) } = \color{blue}{ 48 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-4}&1&-8&12&12&-38\\& & -4& \color{blue}{48} & & \\ \hline &1&\color{blue}{-12}&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 12 } + \color{orangered}{ 48 } = \color{orangered}{ 60 } $
$$ \begin{array}{c|rrrrr}-4&1&-8&\color{orangered}{ 12 }&12&-38\\& & -4& \color{orangered}{48} & & \\ \hline &1&-12&\color{orangered}{60}&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -4 } \cdot \color{blue}{ 60 } = \color{blue}{ -240 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-4}&1&-8&12&12&-38\\& & -4& 48& \color{blue}{-240} & \\ \hline &1&-12&\color{blue}{60}&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 12 } + \color{orangered}{ \left( -240 \right) } = \color{orangered}{ -228 } $
$$ \begin{array}{c|rrrrr}-4&1&-8&12&\color{orangered}{ 12 }&-38\\& & -4& 48& \color{orangered}{-240} & \\ \hline &1&-12&60&\color{orangered}{-228}& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -4 } \cdot \color{blue}{ \left( -228 \right) } = \color{blue}{ 912 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-4}&1&-8&12&12&-38\\& & -4& 48& -240& \color{blue}{912} \\ \hline &1&-12&60&\color{blue}{-228}& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ -38 } + \color{orangered}{ 912 } = \color{orangered}{ 874 } $
$$ \begin{array}{c|rrrrr}-4&1&-8&12&12&\color{orangered}{ -38 }\\& & -4& 48& -240& \color{orangered}{912} \\ \hline &\color{blue}{1}&\color{blue}{-12}&\color{blue}{60}&\color{blue}{-228}&\color{orangered}{874} \end{array} $$Bottom line represents the quotient $ \color{blue}{ x^{3}-12x^{2}+60x-228 } $ with a remainder of $ \color{red}{ 874 } $.