Divide using synthetic division $$ ( -x^{5}-13x^{4}-120x+80 ) \div ( x+3 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrrr}-3&-1&-13&0&0&-120&80\\& & 3& 30& -90& 270& \color{black}{-450} \\ \hline &\color{blue}{-1}&\color{blue}{-10}&\color{blue}{30}&\color{blue}{-90}&\color{blue}{150}&\color{orangered}{-370} \end{array} $$The solution is:
$$ \frac{ -x^{5}-13x^{4}-120x+80 }{ x+3 } = \color{blue}{-x^{4}-10x^{3}+30x^{2}-90x+150} \color{red}{~-~} \frac{ \color{red}{ 370 } }{ x+3 } $$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.
$$ \begin{array}{c|rrrrrr}\color{blue}{-3}&-1&-13&0&0&-120&80\\& & & & & & \\ \hline &&&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrrr}-3&\color{orangered}{ -1 }&-13&0&0&-120&80\\& & & & & & \\ \hline &\color{orangered}{-1}&&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -3 } \cdot \color{blue}{ \left( -1 \right) } = \color{blue}{ 3 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{-3}&-1&-13&0&0&-120&80\\& & \color{blue}{3} & & & & \\ \hline &\color{blue}{-1}&&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -13 } + \color{orangered}{ 3 } = \color{orangered}{ -10 } $
$$ \begin{array}{c|rrrrrr}-3&-1&\color{orangered}{ -13 }&0&0&-120&80\\& & \color{orangered}{3} & & & & \\ \hline &-1&\color{orangered}{-10}&&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -3 } \cdot \color{blue}{ \left( -10 \right) } = \color{blue}{ 30 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{-3}&-1&-13&0&0&-120&80\\& & 3& \color{blue}{30} & & & \\ \hline &-1&\color{blue}{-10}&&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 30 } = \color{orangered}{ 30 } $
$$ \begin{array}{c|rrrrrr}-3&-1&-13&\color{orangered}{ 0 }&0&-120&80\\& & 3& \color{orangered}{30} & & & \\ \hline &-1&-10&\color{orangered}{30}&&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -3 } \cdot \color{blue}{ 30 } = \color{blue}{ -90 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{-3}&-1&-13&0&0&-120&80\\& & 3& 30& \color{blue}{-90} & & \\ \hline &-1&-10&\color{blue}{30}&&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ \left( -90 \right) } = \color{orangered}{ -90 } $
$$ \begin{array}{c|rrrrrr}-3&-1&-13&0&\color{orangered}{ 0 }&-120&80\\& & 3& 30& \color{orangered}{-90} & & \\ \hline &-1&-10&30&\color{orangered}{-90}&& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -3 } \cdot \color{blue}{ \left( -90 \right) } = \color{blue}{ 270 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{-3}&-1&-13&0&0&-120&80\\& & 3& 30& -90& \color{blue}{270} & \\ \hline &-1&-10&30&\color{blue}{-90}&& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ -120 } + \color{orangered}{ 270 } = \color{orangered}{ 150 } $
$$ \begin{array}{c|rrrrrr}-3&-1&-13&0&0&\color{orangered}{ -120 }&80\\& & 3& 30& -90& \color{orangered}{270} & \\ \hline &-1&-10&30&-90&\color{orangered}{150}& \end{array} $$Step 10 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -3 } \cdot \color{blue}{ 150 } = \color{blue}{ -450 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{-3}&-1&-13&0&0&-120&80\\& & 3& 30& -90& 270& \color{blue}{-450} \\ \hline &-1&-10&30&-90&\color{blue}{150}& \end{array} $$Step 11 : Add down last column: $ \color{orangered}{ 80 } + \color{orangered}{ \left( -450 \right) } = \color{orangered}{ -370 } $
$$ \begin{array}{c|rrrrrr}-3&-1&-13&0&0&-120&\color{orangered}{ 80 }\\& & 3& 30& -90& 270& \color{orangered}{-450} \\ \hline &\color{blue}{-1}&\color{blue}{-10}&\color{blue}{30}&\color{blue}{-90}&\color{blue}{150}&\color{orangered}{-370} \end{array} $$Bottom line represents the quotient $ \color{blue}{ -x^{4}-10x^{3}+30x^{2}-90x+150 } $ with a remainder of $ \color{red}{ -370 } $.