Divide using synthetic division $$ ( 23x^{5}-33x^{4}-32x^{3}+11x^{2}+15x-36 ) \div ( x-2 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrrr}2&23&-33&-32&11&15&-36\\& & 46& 26& -12& -2& \color{black}{26} \\ \hline &\color{blue}{23}&\color{blue}{13}&\color{blue}{-6}&\color{blue}{-1}&\color{blue}{13}&\color{orangered}{-10} \end{array} $$The solution is:
$$ \frac{ 23x^{5}-33x^{4}-32x^{3}+11x^{2}+15x-36 }{ x-2 } = \color{blue}{23x^{4}+13x^{3}-6x^{2}-x+13} \color{red}{~-~} \frac{ \color{red}{ 10 } }{ x-2 } $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x -2 = 0 $ ( $ x = \color{blue}{ 2 } $ ) at the left.
$$ \begin{array}{c|rrrrrr}\color{blue}{2}&23&-33&-32&11&15&-36\\& & & & & & \\ \hline &&&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrrr}2&\color{orangered}{ 23 }&-33&-32&11&15&-36\\& & & & & & \\ \hline &\color{orangered}{23}&&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ 23 } = \color{blue}{ 46 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{2}&23&-33&-32&11&15&-36\\& & \color{blue}{46} & & & & \\ \hline &\color{blue}{23}&&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -33 } + \color{orangered}{ 46 } = \color{orangered}{ 13 } $
$$ \begin{array}{c|rrrrrr}2&23&\color{orangered}{ -33 }&-32&11&15&-36\\& & \color{orangered}{46} & & & & \\ \hline &23&\color{orangered}{13}&&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ 13 } = \color{blue}{ 26 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{2}&23&-33&-32&11&15&-36\\& & 46& \color{blue}{26} & & & \\ \hline &23&\color{blue}{13}&&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -32 } + \color{orangered}{ 26 } = \color{orangered}{ -6 } $
$$ \begin{array}{c|rrrrrr}2&23&-33&\color{orangered}{ -32 }&11&15&-36\\& & 46& \color{orangered}{26} & & & \\ \hline &23&13&\color{orangered}{-6}&&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ \left( -6 \right) } = \color{blue}{ -12 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{2}&23&-33&-32&11&15&-36\\& & 46& 26& \color{blue}{-12} & & \\ \hline &23&13&\color{blue}{-6}&&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 11 } + \color{orangered}{ \left( -12 \right) } = \color{orangered}{ -1 } $
$$ \begin{array}{c|rrrrrr}2&23&-33&-32&\color{orangered}{ 11 }&15&-36\\& & 46& 26& \color{orangered}{-12} & & \\ \hline &23&13&-6&\color{orangered}{-1}&& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ \left( -1 \right) } = \color{blue}{ -2 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{2}&23&-33&-32&11&15&-36\\& & 46& 26& -12& \color{blue}{-2} & \\ \hline &23&13&-6&\color{blue}{-1}&& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ 15 } + \color{orangered}{ \left( -2 \right) } = \color{orangered}{ 13 } $
$$ \begin{array}{c|rrrrrr}2&23&-33&-32&11&\color{orangered}{ 15 }&-36\\& & 46& 26& -12& \color{orangered}{-2} & \\ \hline &23&13&-6&-1&\color{orangered}{13}& \end{array} $$Step 10 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ 13 } = \color{blue}{ 26 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{2}&23&-33&-32&11&15&-36\\& & 46& 26& -12& -2& \color{blue}{26} \\ \hline &23&13&-6&-1&\color{blue}{13}& \end{array} $$Step 11 : Add down last column: $ \color{orangered}{ -36 } + \color{orangered}{ 26 } = \color{orangered}{ -10 } $
$$ \begin{array}{c|rrrrrr}2&23&-33&-32&11&15&\color{orangered}{ -36 }\\& & 46& 26& -12& -2& \color{orangered}{26} \\ \hline &\color{blue}{23}&\color{blue}{13}&\color{blue}{-6}&\color{blue}{-1}&\color{blue}{13}&\color{orangered}{-10} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 23x^{4}+13x^{3}-6x^{2}-x+13 } $ with a remainder of $ \color{red}{ -10 } $.