Divide using synthetic division $$ ( 2x^{5}+4x^{4}-2x^{2}-5x+9 ) \div ( x-3 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrrr}3&2&4&0&-2&-5&9\\& & 6& 30& 90& 264& \color{black}{777} \\ \hline &\color{blue}{2}&\color{blue}{10}&\color{blue}{30}&\color{blue}{88}&\color{blue}{259}&\color{orangered}{786} \end{array} $$The solution is:
$$ \frac{ 2x^{5}+4x^{4}-2x^{2}-5x+9 }{ x-3 } = \color{blue}{2x^{4}+10x^{3}+30x^{2}+88x+259} ~+~ \frac{ \color{red}{ 786 } }{ x-3 } $$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.
$$ \begin{array}{c|rrrrrr}\color{blue}{3}&2&4&0&-2&-5&9\\& & & & & & \\ \hline &&&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrrr}3&\color{orangered}{ 2 }&4&0&-2&-5&9\\& & & & & & \\ \hline &\color{orangered}{2}&&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 3 } \cdot \color{blue}{ 2 } = \color{blue}{ 6 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{3}&2&4&0&-2&-5&9\\& & \color{blue}{6} & & & & \\ \hline &\color{blue}{2}&&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 4 } + \color{orangered}{ 6 } = \color{orangered}{ 10 } $
$$ \begin{array}{c|rrrrrr}3&2&\color{orangered}{ 4 }&0&-2&-5&9\\& & \color{orangered}{6} & & & & \\ \hline &2&\color{orangered}{10}&&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 3 } \cdot \color{blue}{ 10 } = \color{blue}{ 30 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{3}&2&4&0&-2&-5&9\\& & 6& \color{blue}{30} & & & \\ \hline &2&\color{blue}{10}&&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 30 } = \color{orangered}{ 30 } $
$$ \begin{array}{c|rrrrrr}3&2&4&\color{orangered}{ 0 }&-2&-5&9\\& & 6& \color{orangered}{30} & & & \\ \hline &2&10&\color{orangered}{30}&&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 3 } \cdot \color{blue}{ 30 } = \color{blue}{ 90 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{3}&2&4&0&-2&-5&9\\& & 6& 30& \color{blue}{90} & & \\ \hline &2&10&\color{blue}{30}&&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -2 } + \color{orangered}{ 90 } = \color{orangered}{ 88 } $
$$ \begin{array}{c|rrrrrr}3&2&4&0&\color{orangered}{ -2 }&-5&9\\& & 6& 30& \color{orangered}{90} & & \\ \hline &2&10&30&\color{orangered}{88}&& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 3 } \cdot \color{blue}{ 88 } = \color{blue}{ 264 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{3}&2&4&0&-2&-5&9\\& & 6& 30& 90& \color{blue}{264} & \\ \hline &2&10&30&\color{blue}{88}&& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ -5 } + \color{orangered}{ 264 } = \color{orangered}{ 259 } $
$$ \begin{array}{c|rrrrrr}3&2&4&0&-2&\color{orangered}{ -5 }&9\\& & 6& 30& 90& \color{orangered}{264} & \\ \hline &2&10&30&88&\color{orangered}{259}& \end{array} $$Step 10 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 3 } \cdot \color{blue}{ 259 } = \color{blue}{ 777 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{3}&2&4&0&-2&-5&9\\& & 6& 30& 90& 264& \color{blue}{777} \\ \hline &2&10&30&88&\color{blue}{259}& \end{array} $$Step 11 : Add down last column: $ \color{orangered}{ 9 } + \color{orangered}{ 777 } = \color{orangered}{ 786 } $
$$ \begin{array}{c|rrrrrr}3&2&4&0&-2&-5&\color{orangered}{ 9 }\\& & 6& 30& 90& 264& \color{orangered}{777} \\ \hline &\color{blue}{2}&\color{blue}{10}&\color{blue}{30}&\color{blue}{88}&\color{blue}{259}&\color{orangered}{786} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 2x^{4}+10x^{3}+30x^{2}+88x+259 } $ with a remainder of $ \color{red}{ 786 } $.