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