Divide using synthetic division $$ ( 3x^{3}+5x^{2}+12x+3 ) \div ( x+3 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}-3&3&5&12&3\\& & -9& 12& \color{black}{-72} \\ \hline &\color{blue}{3}&\color{blue}{-4}&\color{blue}{24}&\color{orangered}{-69} \end{array} $$The solution is:
$$ \frac{ 3x^{3}+5x^{2}+12x+3 }{ x+3 } = \color{blue}{3x^{2}-4x+24} \color{red}{~-~} \frac{ \color{red}{ 69 } }{ x+3 } $$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.
$$ \begin{array}{c|rrrr}\color{blue}{-3}&3&5&12&3\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}-3&\color{orangered}{ 3 }&5&12&3\\& & & & \\ \hline &\color{orangered}{3}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -3 } \cdot \color{blue}{ 3 } = \color{blue}{ -9 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-3}&3&5&12&3\\& & \color{blue}{-9} & & \\ \hline &\color{blue}{3}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 5 } + \color{orangered}{ \left( -9 \right) } = \color{orangered}{ -4 } $
$$ \begin{array}{c|rrrr}-3&3&\color{orangered}{ 5 }&12&3\\& & \color{orangered}{-9} & & \\ \hline &3&\color{orangered}{-4}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -3 } \cdot \color{blue}{ \left( -4 \right) } = \color{blue}{ 12 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-3}&3&5&12&3\\& & -9& \color{blue}{12} & \\ \hline &3&\color{blue}{-4}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 12 } + \color{orangered}{ 12 } = \color{orangered}{ 24 } $
$$ \begin{array}{c|rrrr}-3&3&5&\color{orangered}{ 12 }&3\\& & -9& \color{orangered}{12} & \\ \hline &3&-4&\color{orangered}{24}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -3 } \cdot \color{blue}{ 24 } = \color{blue}{ -72 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-3}&3&5&12&3\\& & -9& 12& \color{blue}{-72} \\ \hline &3&-4&\color{blue}{24}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 3 } + \color{orangered}{ \left( -72 \right) } = \color{orangered}{ -69 } $
$$ \begin{array}{c|rrrr}-3&3&5&12&\color{orangered}{ 3 }\\& & -9& 12& \color{orangered}{-72} \\ \hline &\color{blue}{3}&\color{blue}{-4}&\color{blue}{24}&\color{orangered}{-69} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 3x^{2}-4x+24 } $ with a remainder of $ \color{red}{ -69 } $.