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