Divide using synthetic division $$ ( x^{3}-8x^{2}+32x+371 ) \div ( x+4 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}-4&1&-8&32&371\\& & -4& 48& \color{black}{-320} \\ \hline &\color{blue}{1}&\color{blue}{-12}&\color{blue}{80}&\color{orangered}{51} \end{array} $$The solution is:
$$ \frac{ x^{3}-8x^{2}+32x+371 }{ x+4 } = \color{blue}{x^{2}-12x+80} ~+~ \frac{ \color{red}{ 51 } }{ x+4 } $$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.
$$ \begin{array}{c|rrrr}\color{blue}{-4}&1&-8&32&371\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}-4&\color{orangered}{ 1 }&-8&32&371\\& & & & \\ \hline &\color{orangered}{1}&&& \end{array} $$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 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-4}&1&-8&32&371\\& & \color{blue}{-4} & & \\ \hline &\color{blue}{1}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -8 } + \color{orangered}{ \left( -4 \right) } = \color{orangered}{ -12 } $
$$ \begin{array}{c|rrrr}-4&1&\color{orangered}{ -8 }&32&371\\& & \color{orangered}{-4} & & \\ \hline &1&\color{orangered}{-12}&& \end{array} $$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 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-4}&1&-8&32&371\\& & -4& \color{blue}{48} & \\ \hline &1&\color{blue}{-12}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 32 } + \color{orangered}{ 48 } = \color{orangered}{ 80 } $
$$ \begin{array}{c|rrrr}-4&1&-8&\color{orangered}{ 32 }&371\\& & -4& \color{orangered}{48} & \\ \hline &1&-12&\color{orangered}{80}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -4 } \cdot \color{blue}{ 80 } = \color{blue}{ -320 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-4}&1&-8&32&371\\& & -4& 48& \color{blue}{-320} \\ \hline &1&-12&\color{blue}{80}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 371 } + \color{orangered}{ \left( -320 \right) } = \color{orangered}{ 51 } $
$$ \begin{array}{c|rrrr}-4&1&-8&32&\color{orangered}{ 371 }\\& & -4& 48& \color{orangered}{-320} \\ \hline &\color{blue}{1}&\color{blue}{-12}&\color{blue}{80}&\color{orangered}{51} \end{array} $$Bottom line represents the quotient $ \color{blue}{ x^{2}-12x+80 } $ with a remainder of $ \color{red}{ 51 } $.