Divide using synthetic division $$ ( 3x^{4}-2x^{3}-x^{2}-5x+3 ) \div ( x+1 ) $$
The synthetic division table is:
$$ \begin{array}{c|rrrrr}-1&3&-2&-1&-5&3\\& & -3& 5& -4& \color{black}{9} \\ \hline &\color{blue}{3}&\color{blue}{-5}&\color{blue}{4}&\color{blue}{-9}&\color{orangered}{12} \end{array} $$The solution is:
$$ \dfrac{ 3x^{4}-2x^{3}-x^{2}-5x+3 }{ x+1 } = \color{blue}{3x^{3}-5x^{2}+4x-9} ~+~ \dfrac{ \color{red}{ 12 } }{ x+1 } $$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.
$$ \begin{array}{c|rrrrr}\color{blue}{-1}&3&-2&-1&-5&3\\& & & & & \\ \hline &&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrr}-1&\color{orangered}{ 3 }&-2&-1&-5&3\\& & & & & \\ \hline &\color{orangered}{3}&&&& \end{array} $$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 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-1}&3&-2&-1&-5&3\\& & \color{blue}{-3} & & & \\ \hline &\color{blue}{3}&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -2 } + \color{orangered}{ \left( -3 \right) } = \color{orangered}{ -5 } $
$$ \begin{array}{c|rrrrr}-1&3&\color{orangered}{ -2 }&-1&-5&3\\& & \color{orangered}{-3} & & & \\ \hline &3&\color{orangered}{-5}&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -1 } \cdot \color{blue}{ \left( -5 \right) } = \color{blue}{ 5 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-1}&3&-2&-1&-5&3\\& & -3& \color{blue}{5} & & \\ \hline &3&\color{blue}{-5}&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -1 } + \color{orangered}{ 5 } = \color{orangered}{ 4 } $
$$ \begin{array}{c|rrrrr}-1&3&-2&\color{orangered}{ -1 }&-5&3\\& & -3& \color{orangered}{5} & & \\ \hline &3&-5&\color{orangered}{4}&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -1 } \cdot \color{blue}{ 4 } = \color{blue}{ -4 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-1}&3&-2&-1&-5&3\\& & -3& 5& \color{blue}{-4} & \\ \hline &3&-5&\color{blue}{4}&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -5 } + \color{orangered}{ \left( -4 \right) } = \color{orangered}{ -9 } $
$$ \begin{array}{c|rrrrr}-1&3&-2&-1&\color{orangered}{ -5 }&3\\& & -3& 5& \color{orangered}{-4} & \\ \hline &3&-5&4&\color{orangered}{-9}& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -1 } \cdot \color{blue}{ \left( -9 \right) } = \color{blue}{ 9 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-1}&3&-2&-1&-5&3\\& & -3& 5& -4& \color{blue}{9} \\ \hline &3&-5&4&\color{blue}{-9}& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ 3 } + \color{orangered}{ 9 } = \color{orangered}{ 12 } $
$$ \begin{array}{c|rrrrr}-1&3&-2&-1&-5&\color{orangered}{ 3 }\\& & -3& 5& -4& \color{orangered}{9} \\ \hline &\color{blue}{3}&\color{blue}{-5}&\color{blue}{4}&\color{blue}{-9}&\color{orangered}{12} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 3x^{3}-5x^{2}+4x-9 } $ with a remainder of $ \color{red}{ 12 } $.