Divide using synthetic division $$ ( 3x^{3}-5x^{2}+2x+3 ) \div ( x+2 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}-2&3&-5&2&3\\& & -6& 22& \color{black}{-48} \\ \hline &\color{blue}{3}&\color{blue}{-11}&\color{blue}{24}&\color{orangered}{-45} \end{array} $$The solution is:
$$ \frac{ 3x^{3}-5x^{2}+2x+3 }{ x+2 } = \color{blue}{3x^{2}-11x+24} \color{red}{~-~} \frac{ \color{red}{ 45 } }{ x+2 } $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x + 2 = 0 $ ( $ x = \color{blue}{ -2 } $ ) at the left.
$$ \begin{array}{c|rrrr}\color{blue}{-2}&3&-5&2&3\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}-2&\color{orangered}{ 3 }&-5&2&3\\& & & & \\ \hline &\color{orangered}{3}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ 3 } = \color{blue}{ -6 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-2}&3&-5&2&3\\& & \color{blue}{-6} & & \\ \hline &\color{blue}{3}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -5 } + \color{orangered}{ \left( -6 \right) } = \color{orangered}{ -11 } $
$$ \begin{array}{c|rrrr}-2&3&\color{orangered}{ -5 }&2&3\\& & \color{orangered}{-6} & & \\ \hline &3&\color{orangered}{-11}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ \left( -11 \right) } = \color{blue}{ 22 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-2}&3&-5&2&3\\& & -6& \color{blue}{22} & \\ \hline &3&\color{blue}{-11}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 2 } + \color{orangered}{ 22 } = \color{orangered}{ 24 } $
$$ \begin{array}{c|rrrr}-2&3&-5&\color{orangered}{ 2 }&3\\& & -6& \color{orangered}{22} & \\ \hline &3&-11&\color{orangered}{24}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ 24 } = \color{blue}{ -48 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-2}&3&-5&2&3\\& & -6& 22& \color{blue}{-48} \\ \hline &3&-11&\color{blue}{24}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 3 } + \color{orangered}{ \left( -48 \right) } = \color{orangered}{ -45 } $
$$ \begin{array}{c|rrrr}-2&3&-5&2&\color{orangered}{ 3 }\\& & -6& 22& \color{orangered}{-48} \\ \hline &\color{blue}{3}&\color{blue}{-11}&\color{blue}{24}&\color{orangered}{-45} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 3x^{2}-11x+24 } $ with a remainder of $ \color{red}{ -45 } $.