Divide using synthetic division $$ ( 8x^{4}+2x^{2}+3x+27 ) \div ( 2x+3 ) $$
The synthetic division table is:
$$ \begin{array}{c|rrrrr}-3&8&0&2&3&27\\& & -24& 72& -222& \color{black}{657} \\ \hline &\color{blue}{8}&\color{blue}{-24}&\color{blue}{74}&\color{blue}{-219}&\color{orangered}{684} \end{array} $$The solution is:
$$ \dfrac{ 8x^{4}+2x^{2}+3x+27 }{ x+3 } = \color{blue}{8x^{3}-24x^{2}+74x-219} ~+~ \dfrac{ \color{red}{ 684 } }{ x+3 } $$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.
$$ \begin{array}{c|rrrrr}\color{blue}{-3}&8&0&2&3&27\\& & & & & \\ \hline &&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrr}-3&\color{orangered}{ 8 }&0&2&3&27\\& & & & & \\ \hline &\color{orangered}{8}&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -3 } \cdot \color{blue}{ 8 } = \color{blue}{ -24 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-3}&8&0&2&3&27\\& & \color{blue}{-24} & & & \\ \hline &\color{blue}{8}&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ \left( -24 \right) } = \color{orangered}{ -24 } $
$$ \begin{array}{c|rrrrr}-3&8&\color{orangered}{ 0 }&2&3&27\\& & \color{orangered}{-24} & & & \\ \hline &8&\color{orangered}{-24}&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -3 } \cdot \color{blue}{ \left( -24 \right) } = \color{blue}{ 72 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-3}&8&0&2&3&27\\& & -24& \color{blue}{72} & & \\ \hline &8&\color{blue}{-24}&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 2 } + \color{orangered}{ 72 } = \color{orangered}{ 74 } $
$$ \begin{array}{c|rrrrr}-3&8&0&\color{orangered}{ 2 }&3&27\\& & -24& \color{orangered}{72} & & \\ \hline &8&-24&\color{orangered}{74}&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -3 } \cdot \color{blue}{ 74 } = \color{blue}{ -222 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-3}&8&0&2&3&27\\& & -24& 72& \color{blue}{-222} & \\ \hline &8&-24&\color{blue}{74}&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 3 } + \color{orangered}{ \left( -222 \right) } = \color{orangered}{ -219 } $
$$ \begin{array}{c|rrrrr}-3&8&0&2&\color{orangered}{ 3 }&27\\& & -24& 72& \color{orangered}{-222} & \\ \hline &8&-24&74&\color{orangered}{-219}& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -3 } \cdot \color{blue}{ \left( -219 \right) } = \color{blue}{ 657 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-3}&8&0&2&3&27\\& & -24& 72& -222& \color{blue}{657} \\ \hline &8&-24&74&\color{blue}{-219}& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ 27 } + \color{orangered}{ 657 } = \color{orangered}{ 684 } $
$$ \begin{array}{c|rrrrr}-3&8&0&2&3&\color{orangered}{ 27 }\\& & -24& 72& -222& \color{orangered}{657} \\ \hline &\color{blue}{8}&\color{blue}{-24}&\color{blue}{74}&\color{blue}{-219}&\color{orangered}{684} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 8x^{3}-24x^{2}+74x-219 } $ with a remainder of $ \color{red}{ 684 } $.