Divide using synthetic division $$ ( 6x^{3}+x^{2}+4x+4 ) \div ( 3x-2 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}2&6&1&4&4\\& & 12& 26& \color{black}{60} \\ \hline &\color{blue}{6}&\color{blue}{13}&\color{blue}{30}&\color{orangered}{64} \end{array} $$The solution is:
$$ \frac{ 6x^{3}+x^{2}+4x+4 }{ x-2 } = \color{blue}{6x^{2}+13x+30} ~+~ \frac{ \color{red}{ 64 } }{ x-2 } $$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.
$$ \begin{array}{c|rrrr}\color{blue}{2}&6&1&4&4\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}2&\color{orangered}{ 6 }&1&4&4\\& & & & \\ \hline &\color{orangered}{6}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ 6 } = \color{blue}{ 12 } $.
$$ \begin{array}{c|rrrr}\color{blue}{2}&6&1&4&4\\& & \color{blue}{12} & & \\ \hline &\color{blue}{6}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 1 } + \color{orangered}{ 12 } = \color{orangered}{ 13 } $
$$ \begin{array}{c|rrrr}2&6&\color{orangered}{ 1 }&4&4\\& & \color{orangered}{12} & & \\ \hline &6&\color{orangered}{13}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ 13 } = \color{blue}{ 26 } $.
$$ \begin{array}{c|rrrr}\color{blue}{2}&6&1&4&4\\& & 12& \color{blue}{26} & \\ \hline &6&\color{blue}{13}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 4 } + \color{orangered}{ 26 } = \color{orangered}{ 30 } $
$$ \begin{array}{c|rrrr}2&6&1&\color{orangered}{ 4 }&4\\& & 12& \color{orangered}{26} & \\ \hline &6&13&\color{orangered}{30}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ 30 } = \color{blue}{ 60 } $.
$$ \begin{array}{c|rrrr}\color{blue}{2}&6&1&4&4\\& & 12& 26& \color{blue}{60} \\ \hline &6&13&\color{blue}{30}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 4 } + \color{orangered}{ 60 } = \color{orangered}{ 64 } $
$$ \begin{array}{c|rrrr}2&6&1&4&\color{orangered}{ 4 }\\& & 12& 26& \color{orangered}{60} \\ \hline &\color{blue}{6}&\color{blue}{13}&\color{blue}{30}&\color{orangered}{64} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 6x^{2}+13x+30 } $ with a remainder of $ \color{red}{ 64 } $.