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