Divide using synthetic division $$ ( 18x^{3}+39x^{2}+8x+16 ) \div ( 3x-4 ) $$
The synthetic division table is:
$$ \begin{array}{c|rrrr}4&18&39&8&16\\& & 72& 444& \color{black}{1808} \\ \hline &\color{blue}{18}&\color{blue}{111}&\color{blue}{452}&\color{orangered}{1824} \end{array} $$The solution is:
$$ \dfrac{ 18x^{3}+39x^{2}+8x+16 }{ x-4 } = \color{blue}{18x^{2}+111x+452} ~+~ \dfrac{ \color{red}{ 1824 } }{ x-4 } $$Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x -4 = 0 $ ( $ x = \color{blue}{ 4 } $ ) at the left.
$$ \begin{array}{c|rrrr}\color{blue}{4}&18&39&8&16\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}4&\color{orangered}{ 18 }&39&8&16\\& & & & \\ \hline &\color{orangered}{18}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 4 } \cdot \color{blue}{ 18 } = \color{blue}{ 72 } $.
$$ \begin{array}{c|rrrr}\color{blue}{4}&18&39&8&16\\& & \color{blue}{72} & & \\ \hline &\color{blue}{18}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 39 } + \color{orangered}{ 72 } = \color{orangered}{ 111 } $
$$ \begin{array}{c|rrrr}4&18&\color{orangered}{ 39 }&8&16\\& & \color{orangered}{72} & & \\ \hline &18&\color{orangered}{111}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 4 } \cdot \color{blue}{ 111 } = \color{blue}{ 444 } $.
$$ \begin{array}{c|rrrr}\color{blue}{4}&18&39&8&16\\& & 72& \color{blue}{444} & \\ \hline &18&\color{blue}{111}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 8 } + \color{orangered}{ 444 } = \color{orangered}{ 452 } $
$$ \begin{array}{c|rrrr}4&18&39&\color{orangered}{ 8 }&16\\& & 72& \color{orangered}{444} & \\ \hline &18&111&\color{orangered}{452}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 4 } \cdot \color{blue}{ 452 } = \color{blue}{ 1808 } $.
$$ \begin{array}{c|rrrr}\color{blue}{4}&18&39&8&16\\& & 72& 444& \color{blue}{1808} \\ \hline &18&111&\color{blue}{452}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 16 } + \color{orangered}{ 1808 } = \color{orangered}{ 1824 } $
$$ \begin{array}{c|rrrr}4&18&39&8&\color{orangered}{ 16 }\\& & 72& 444& \color{orangered}{1808} \\ \hline &\color{blue}{18}&\color{blue}{111}&\color{blue}{452}&\color{orangered}{1824} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 18x^{2}+111x+452 } $ with a remainder of $ \color{red}{ 1824 } $.