Divide using synthetic division $$ ( x^{3}+3x^{2}-34x-80 ) \div ( x+4 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}-4&1&3&-34&-80\\& & -4& 4& \color{black}{120} \\ \hline &\color{blue}{1}&\color{blue}{-1}&\color{blue}{-30}&\color{orangered}{40} \end{array} $$The solution is:
$$ \frac{ x^{3}+3x^{2}-34x-80 }{ x+4 } = \color{blue}{x^{2}-x-30} ~+~ \frac{ \color{red}{ 40 } }{ x+4 } $$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.
$$ \begin{array}{c|rrrr}\color{blue}{-4}&1&3&-34&-80\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}-4&\color{orangered}{ 1 }&3&-34&-80\\& & & & \\ \hline &\color{orangered}{1}&&& \end{array} $$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 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-4}&1&3&-34&-80\\& & \color{blue}{-4} & & \\ \hline &\color{blue}{1}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 3 } + \color{orangered}{ \left( -4 \right) } = \color{orangered}{ -1 } $
$$ \begin{array}{c|rrrr}-4&1&\color{orangered}{ 3 }&-34&-80\\& & \color{orangered}{-4} & & \\ \hline &1&\color{orangered}{-1}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -4 } \cdot \color{blue}{ \left( -1 \right) } = \color{blue}{ 4 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-4}&1&3&-34&-80\\& & -4& \color{blue}{4} & \\ \hline &1&\color{blue}{-1}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -34 } + \color{orangered}{ 4 } = \color{orangered}{ -30 } $
$$ \begin{array}{c|rrrr}-4&1&3&\color{orangered}{ -34 }&-80\\& & -4& \color{orangered}{4} & \\ \hline &1&-1&\color{orangered}{-30}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -4 } \cdot \color{blue}{ \left( -30 \right) } = \color{blue}{ 120 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-4}&1&3&-34&-80\\& & -4& 4& \color{blue}{120} \\ \hline &1&-1&\color{blue}{-30}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -80 } + \color{orangered}{ 120 } = \color{orangered}{ 40 } $
$$ \begin{array}{c|rrrr}-4&1&3&-34&\color{orangered}{ -80 }\\& & -4& 4& \color{orangered}{120} \\ \hline &\color{blue}{1}&\color{blue}{-1}&\color{blue}{-30}&\color{orangered}{40} \end{array} $$Bottom line represents the quotient $ \color{blue}{ x^{2}-x-30 } $ with a remainder of $ \color{red}{ 40 } $.