Divide using synthetic division $$ ( 3x^{8}+2x^{4}+5x^{3}+5x+12 ) \div ( 2x+3 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrrrrrr}-3&3&0&0&0&2&5&0&5&12\\& & -9& 27& -81& 243& -735& 2190& -6570& \color{black}{19695} \\ \hline &\color{blue}{3}&\color{blue}{-9}&\color{blue}{27}&\color{blue}{-81}&\color{blue}{245}&\color{blue}{-730}&\color{blue}{2190}&\color{blue}{-6565}&\color{orangered}{19707} \end{array} $$The solution is:
$$ \frac{ 3x^{8}+2x^{4}+5x^{3}+5x+12 }{ x+3 } = \color{blue}{3x^{7}-9x^{6}+27x^{5}-81x^{4}+245x^{3}-730x^{2}+2190x-6565} ~+~ \frac{ \color{red}{ 19707 } }{ x+3 } $$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.
$$ \begin{array}{c|rrrrrrrrr}\color{blue}{-3}&3&0&0&0&2&5&0&5&12\\& & & & & & & & & \\ \hline &&&&&&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrrrrrr}-3&\color{orangered}{ 3 }&0&0&0&2&5&0&5&12\\& & & & & & & & & \\ \hline &\color{orangered}{3}&&&&&&&& \end{array} $$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 } $.
$$ \begin{array}{c|rrrrrrrrr}\color{blue}{-3}&3&0&0&0&2&5&0&5&12\\& & \color{blue}{-9} & & & & & & & \\ \hline &\color{blue}{3}&&&&&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ \left( -9 \right) } = \color{orangered}{ -9 } $
$$ \begin{array}{c|rrrrrrrrr}-3&3&\color{orangered}{ 0 }&0&0&2&5&0&5&12\\& & \color{orangered}{-9} & & & & & & & \\ \hline &3&\color{orangered}{-9}&&&&&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -3 } \cdot \color{blue}{ \left( -9 \right) } = \color{blue}{ 27 } $.
$$ \begin{array}{c|rrrrrrrrr}\color{blue}{-3}&3&0&0&0&2&5&0&5&12\\& & -9& \color{blue}{27} & & & & & & \\ \hline &3&\color{blue}{-9}&&&&&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 27 } = \color{orangered}{ 27 } $
$$ \begin{array}{c|rrrrrrrrr}-3&3&0&\color{orangered}{ 0 }&0&2&5&0&5&12\\& & -9& \color{orangered}{27} & & & & & & \\ \hline &3&-9&\color{orangered}{27}&&&&&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -3 } \cdot \color{blue}{ 27 } = \color{blue}{ -81 } $.
$$ \begin{array}{c|rrrrrrrrr}\color{blue}{-3}&3&0&0&0&2&5&0&5&12\\& & -9& 27& \color{blue}{-81} & & & & & \\ \hline &3&-9&\color{blue}{27}&&&&&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ \left( -81 \right) } = \color{orangered}{ -81 } $
$$ \begin{array}{c|rrrrrrrrr}-3&3&0&0&\color{orangered}{ 0 }&2&5&0&5&12\\& & -9& 27& \color{orangered}{-81} & & & & & \\ \hline &3&-9&27&\color{orangered}{-81}&&&&& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -3 } \cdot \color{blue}{ \left( -81 \right) } = \color{blue}{ 243 } $.
$$ \begin{array}{c|rrrrrrrrr}\color{blue}{-3}&3&0&0&0&2&5&0&5&12\\& & -9& 27& -81& \color{blue}{243} & & & & \\ \hline &3&-9&27&\color{blue}{-81}&&&&& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ 2 } + \color{orangered}{ 243 } = \color{orangered}{ 245 } $
$$ \begin{array}{c|rrrrrrrrr}-3&3&0&0&0&\color{orangered}{ 2 }&5&0&5&12\\& & -9& 27& -81& \color{orangered}{243} & & & & \\ \hline &3&-9&27&-81&\color{orangered}{245}&&&& \end{array} $$Step 10 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -3 } \cdot \color{blue}{ 245 } = \color{blue}{ -735 } $.
$$ \begin{array}{c|rrrrrrrrr}\color{blue}{-3}&3&0&0&0&2&5&0&5&12\\& & -9& 27& -81& 243& \color{blue}{-735} & & & \\ \hline &3&-9&27&-81&\color{blue}{245}&&&& \end{array} $$Step 11 : Add down last column: $ \color{orangered}{ 5 } + \color{orangered}{ \left( -735 \right) } = \color{orangered}{ -730 } $
$$ \begin{array}{c|rrrrrrrrr}-3&3&0&0&0&2&\color{orangered}{ 5 }&0&5&12\\& & -9& 27& -81& 243& \color{orangered}{-735} & & & \\ \hline &3&-9&27&-81&245&\color{orangered}{-730}&&& \end{array} $$Step 12 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -3 } \cdot \color{blue}{ \left( -730 \right) } = \color{blue}{ 2190 } $.
$$ \begin{array}{c|rrrrrrrrr}\color{blue}{-3}&3&0&0&0&2&5&0&5&12\\& & -9& 27& -81& 243& -735& \color{blue}{2190} & & \\ \hline &3&-9&27&-81&245&\color{blue}{-730}&&& \end{array} $$Step 13 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 2190 } = \color{orangered}{ 2190 } $
$$ \begin{array}{c|rrrrrrrrr}-3&3&0&0&0&2&5&\color{orangered}{ 0 }&5&12\\& & -9& 27& -81& 243& -735& \color{orangered}{2190} & & \\ \hline &3&-9&27&-81&245&-730&\color{orangered}{2190}&& \end{array} $$Step 14 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -3 } \cdot \color{blue}{ 2190 } = \color{blue}{ -6570 } $.
$$ \begin{array}{c|rrrrrrrrr}\color{blue}{-3}&3&0&0&0&2&5&0&5&12\\& & -9& 27& -81& 243& -735& 2190& \color{blue}{-6570} & \\ \hline &3&-9&27&-81&245&-730&\color{blue}{2190}&& \end{array} $$Step 15 : Add down last column: $ \color{orangered}{ 5 } + \color{orangered}{ \left( -6570 \right) } = \color{orangered}{ -6565 } $
$$ \begin{array}{c|rrrrrrrrr}-3&3&0&0&0&2&5&0&\color{orangered}{ 5 }&12\\& & -9& 27& -81& 243& -735& 2190& \color{orangered}{-6570} & \\ \hline &3&-9&27&-81&245&-730&2190&\color{orangered}{-6565}& \end{array} $$Step 16 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -3 } \cdot \color{blue}{ \left( -6565 \right) } = \color{blue}{ 19695 } $.
$$ \begin{array}{c|rrrrrrrrr}\color{blue}{-3}&3&0&0&0&2&5&0&5&12\\& & -9& 27& -81& 243& -735& 2190& -6570& \color{blue}{19695} \\ \hline &3&-9&27&-81&245&-730&2190&\color{blue}{-6565}& \end{array} $$Step 17 : Add down last column: $ \color{orangered}{ 12 } + \color{orangered}{ 19695 } = \color{orangered}{ 19707 } $
$$ \begin{array}{c|rrrrrrrrr}-3&3&0&0&0&2&5&0&5&\color{orangered}{ 12 }\\& & -9& 27& -81& 243& -735& 2190& -6570& \color{orangered}{19695} \\ \hline &\color{blue}{3}&\color{blue}{-9}&\color{blue}{27}&\color{blue}{-81}&\color{blue}{245}&\color{blue}{-730}&\color{blue}{2190}&\color{blue}{-6565}&\color{orangered}{19707} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 3x^{7}-9x^{6}+27x^{5}-81x^{4}+245x^{3}-730x^{2}+2190x-6565 } $ with a remainder of $ \color{red}{ 19707 } $.