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