Divide using synthetic division $$ ( 3x^{4}+14x^{3}+9x^{2}-12x+4 ) \div ( x-3 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrr}3&3&14&9&-12&4\\& & 9& 69& 234& \color{black}{666} \\ \hline &\color{blue}{3}&\color{blue}{23}&\color{blue}{78}&\color{blue}{222}&\color{orangered}{670} \end{array} $$The solution is:
$$ \frac{ 3x^{4}+14x^{3}+9x^{2}-12x+4 }{ x-3 } = \color{blue}{3x^{3}+23x^{2}+78x+222} ~+~ \frac{ \color{red}{ 670 } }{ x-3 } $$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.
$$ \begin{array}{c|rrrrr}\color{blue}{3}&3&14&9&-12&4\\& & & & & \\ \hline &&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrr}3&\color{orangered}{ 3 }&14&9&-12&4\\& & & & & \\ \hline &\color{orangered}{3}&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 3 } \cdot \color{blue}{ 3 } = \color{blue}{ 9 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{3}&3&14&9&-12&4\\& & \color{blue}{9} & & & \\ \hline &\color{blue}{3}&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 14 } + \color{orangered}{ 9 } = \color{orangered}{ 23 } $
$$ \begin{array}{c|rrrrr}3&3&\color{orangered}{ 14 }&9&-12&4\\& & \color{orangered}{9} & & & \\ \hline &3&\color{orangered}{23}&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 3 } \cdot \color{blue}{ 23 } = \color{blue}{ 69 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{3}&3&14&9&-12&4\\& & 9& \color{blue}{69} & & \\ \hline &3&\color{blue}{23}&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 9 } + \color{orangered}{ 69 } = \color{orangered}{ 78 } $
$$ \begin{array}{c|rrrrr}3&3&14&\color{orangered}{ 9 }&-12&4\\& & 9& \color{orangered}{69} & & \\ \hline &3&23&\color{orangered}{78}&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 3 } \cdot \color{blue}{ 78 } = \color{blue}{ 234 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{3}&3&14&9&-12&4\\& & 9& 69& \color{blue}{234} & \\ \hline &3&23&\color{blue}{78}&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -12 } + \color{orangered}{ 234 } = \color{orangered}{ 222 } $
$$ \begin{array}{c|rrrrr}3&3&14&9&\color{orangered}{ -12 }&4\\& & 9& 69& \color{orangered}{234} & \\ \hline &3&23&78&\color{orangered}{222}& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 3 } \cdot \color{blue}{ 222 } = \color{blue}{ 666 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{3}&3&14&9&-12&4\\& & 9& 69& 234& \color{blue}{666} \\ \hline &3&23&78&\color{blue}{222}& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ 4 } + \color{orangered}{ 666 } = \color{orangered}{ 670 } $
$$ \begin{array}{c|rrrrr}3&3&14&9&-12&\color{orangered}{ 4 }\\& & 9& 69& 234& \color{orangered}{666} \\ \hline &\color{blue}{3}&\color{blue}{23}&\color{blue}{78}&\color{blue}{222}&\color{orangered}{670} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 3x^{3}+23x^{2}+78x+222 } $ with a remainder of $ \color{red}{ 670 } $.