Divide using synthetic division $$ ( x^{3}+3x^{2}+4x+3 ) \div ( 3x+6 ) $$
The synthetic division table is:
$$ \begin{array}{c|rrrr}-6&1&3&4&3\\& & -6& 18& \color{black}{-132} \\ \hline &\color{blue}{1}&\color{blue}{-3}&\color{blue}{22}&\color{orangered}{-129} \end{array} $$The solution is:
$$ \dfrac{ x^{3}+3x^{2}+4x+3 }{ x+6 } = \color{blue}{x^{2}-3x+22} \color{red}{~-~} \dfrac{ \color{red}{ 129 } }{ x+6 } $$Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x + 6 = 0 $ ( $ x = \color{blue}{ -6 } $ ) at the left.
$$ \begin{array}{c|rrrr}\color{blue}{-6}&1&3&4&3\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}-6&\color{orangered}{ 1 }&3&4&3\\& & & & \\ \hline &\color{orangered}{1}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -6 } \cdot \color{blue}{ 1 } = \color{blue}{ -6 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-6}&1&3&4&3\\& & \color{blue}{-6} & & \\ \hline &\color{blue}{1}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 3 } + \color{orangered}{ \left( -6 \right) } = \color{orangered}{ -3 } $
$$ \begin{array}{c|rrrr}-6&1&\color{orangered}{ 3 }&4&3\\& & \color{orangered}{-6} & & \\ \hline &1&\color{orangered}{-3}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -6 } \cdot \color{blue}{ \left( -3 \right) } = \color{blue}{ 18 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-6}&1&3&4&3\\& & -6& \color{blue}{18} & \\ \hline &1&\color{blue}{-3}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 4 } + \color{orangered}{ 18 } = \color{orangered}{ 22 } $
$$ \begin{array}{c|rrrr}-6&1&3&\color{orangered}{ 4 }&3\\& & -6& \color{orangered}{18} & \\ \hline &1&-3&\color{orangered}{22}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -6 } \cdot \color{blue}{ 22 } = \color{blue}{ -132 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-6}&1&3&4&3\\& & -6& 18& \color{blue}{-132} \\ \hline &1&-3&\color{blue}{22}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 3 } + \color{orangered}{ \left( -132 \right) } = \color{orangered}{ -129 } $
$$ \begin{array}{c|rrrr}-6&1&3&4&\color{orangered}{ 3 }\\& & -6& 18& \color{orangered}{-132} \\ \hline &\color{blue}{1}&\color{blue}{-3}&\color{blue}{22}&\color{orangered}{-129} \end{array} $$Bottom line represents the quotient $ \color{blue}{ x^{2}-3x+22 } $ with a remainder of $ \color{red}{ -129 } $.