Divide using synthetic division $$ ( x^{4}-13x^{3}-120x+80 ) \div ( x+2 ) $$
The synthetic division table is:
$$ \begin{array}{c|rrrrr}-2&1&-13&0&-120&80\\& & -2& 30& -60& \color{black}{360} \\ \hline &\color{blue}{1}&\color{blue}{-15}&\color{blue}{30}&\color{blue}{-180}&\color{orangered}{440} \end{array} $$The solution is:
$$ \dfrac{ x^{4}-13x^{3}-120x+80 }{ x+2 } = \color{blue}{x^{3}-15x^{2}+30x-180} ~+~ \dfrac{ \color{red}{ 440 } }{ x+2 } $$Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x + 2 = 0 $ ( $ x = \color{blue}{ -2 } $ ) at the left.
$$ \begin{array}{c|rrrrr}\color{blue}{-2}&1&-13&0&-120&80\\& & & & & \\ \hline &&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrr}-2&\color{orangered}{ 1 }&-13&0&-120&80\\& & & & & \\ \hline &\color{orangered}{1}&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ 1 } = \color{blue}{ -2 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-2}&1&-13&0&-120&80\\& & \color{blue}{-2} & & & \\ \hline &\color{blue}{1}&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -13 } + \color{orangered}{ \left( -2 \right) } = \color{orangered}{ -15 } $
$$ \begin{array}{c|rrrrr}-2&1&\color{orangered}{ -13 }&0&-120&80\\& & \color{orangered}{-2} & & & \\ \hline &1&\color{orangered}{-15}&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ \left( -15 \right) } = \color{blue}{ 30 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-2}&1&-13&0&-120&80\\& & -2& \color{blue}{30} & & \\ \hline &1&\color{blue}{-15}&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 30 } = \color{orangered}{ 30 } $
$$ \begin{array}{c|rrrrr}-2&1&-13&\color{orangered}{ 0 }&-120&80\\& & -2& \color{orangered}{30} & & \\ \hline &1&-15&\color{orangered}{30}&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ 30 } = \color{blue}{ -60 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-2}&1&-13&0&-120&80\\& & -2& 30& \color{blue}{-60} & \\ \hline &1&-15&\color{blue}{30}&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -120 } + \color{orangered}{ \left( -60 \right) } = \color{orangered}{ -180 } $
$$ \begin{array}{c|rrrrr}-2&1&-13&0&\color{orangered}{ -120 }&80\\& & -2& 30& \color{orangered}{-60} & \\ \hline &1&-15&30&\color{orangered}{-180}& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ \left( -180 \right) } = \color{blue}{ 360 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-2}&1&-13&0&-120&80\\& & -2& 30& -60& \color{blue}{360} \\ \hline &1&-15&30&\color{blue}{-180}& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ 80 } + \color{orangered}{ 360 } = \color{orangered}{ 440 } $
$$ \begin{array}{c|rrrrr}-2&1&-13&0&-120&\color{orangered}{ 80 }\\& & -2& 30& -60& \color{orangered}{360} \\ \hline &\color{blue}{1}&\color{blue}{-15}&\color{blue}{30}&\color{blue}{-180}&\color{orangered}{440} \end{array} $$Bottom line represents the quotient $ \color{blue}{ x^{3}-15x^{2}+30x-180 } $ with a remainder of $ \color{red}{ 440 } $.