Divide using synthetic division $$ ( 2x^{3}-5x^{2}-x+3 ) \div ( x+3 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}-3&2&-5&-1&3\\& & -6& 33& \color{black}{-96} \\ \hline &\color{blue}{2}&\color{blue}{-11}&\color{blue}{32}&\color{orangered}{-93} \end{array} $$The solution is:
$$ \frac{ 2x^{3}-5x^{2}-x+3 }{ x+3 } = \color{blue}{2x^{2}-11x+32} \color{red}{~-~} \frac{ \color{red}{ 93 } }{ x+3 } $$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.
$$ \begin{array}{c|rrrr}\color{blue}{-3}&2&-5&-1&3\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}-3&\color{orangered}{ 2 }&-5&-1&3\\& & & & \\ \hline &\color{orangered}{2}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -3 } \cdot \color{blue}{ 2 } = \color{blue}{ -6 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-3}&2&-5&-1&3\\& & \color{blue}{-6} & & \\ \hline &\color{blue}{2}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -5 } + \color{orangered}{ \left( -6 \right) } = \color{orangered}{ -11 } $
$$ \begin{array}{c|rrrr}-3&2&\color{orangered}{ -5 }&-1&3\\& & \color{orangered}{-6} & & \\ \hline &2&\color{orangered}{-11}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -3 } \cdot \color{blue}{ \left( -11 \right) } = \color{blue}{ 33 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-3}&2&-5&-1&3\\& & -6& \color{blue}{33} & \\ \hline &2&\color{blue}{-11}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -1 } + \color{orangered}{ 33 } = \color{orangered}{ 32 } $
$$ \begin{array}{c|rrrr}-3&2&-5&\color{orangered}{ -1 }&3\\& & -6& \color{orangered}{33} & \\ \hline &2&-11&\color{orangered}{32}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -3 } \cdot \color{blue}{ 32 } = \color{blue}{ -96 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-3}&2&-5&-1&3\\& & -6& 33& \color{blue}{-96} \\ \hline &2&-11&\color{blue}{32}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 3 } + \color{orangered}{ \left( -96 \right) } = \color{orangered}{ -93 } $
$$ \begin{array}{c|rrrr}-3&2&-5&-1&\color{orangered}{ 3 }\\& & -6& 33& \color{orangered}{-96} \\ \hline &\color{blue}{2}&\color{blue}{-11}&\color{blue}{32}&\color{orangered}{-93} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 2x^{2}-11x+32 } $ with a remainder of $ \color{red}{ -93 } $.