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