Divide using synthetic division $$ ( x^{4}-64x^{2}+2x-9 ) \div ( x-8 ) $$
The synthetic division table is:
$$ \begin{array}{c|rrrrr}8&1&0&-64&2&-9\\& & 8& 64& 0& \color{black}{16} \\ \hline &\color{blue}{1}&\color{blue}{8}&\color{blue}{0}&\color{blue}{2}&\color{orangered}{7} \end{array} $$The solution is:
$$ \dfrac{ x^{4}-64x^{2}+2x-9 }{ x-8 } = \color{blue}{x^{3}+8x^{2}+2} ~+~ \dfrac{ \color{red}{ 7 } }{ x-8 } $$Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x -8 = 0 $ ( $ x = \color{blue}{ 8 } $ ) at the left.
$$ \begin{array}{c|rrrrr}\color{blue}{8}&1&0&-64&2&-9\\& & & & & \\ \hline &&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrr}8&\color{orangered}{ 1 }&0&-64&2&-9\\& & & & & \\ \hline &\color{orangered}{1}&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 8 } \cdot \color{blue}{ 1 } = \color{blue}{ 8 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{8}&1&0&-64&2&-9\\& & \color{blue}{8} & & & \\ \hline &\color{blue}{1}&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 8 } = \color{orangered}{ 8 } $
$$ \begin{array}{c|rrrrr}8&1&\color{orangered}{ 0 }&-64&2&-9\\& & \color{orangered}{8} & & & \\ \hline &1&\color{orangered}{8}&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 8 } \cdot \color{blue}{ 8 } = \color{blue}{ 64 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{8}&1&0&-64&2&-9\\& & 8& \color{blue}{64} & & \\ \hline &1&\color{blue}{8}&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -64 } + \color{orangered}{ 64 } = \color{orangered}{ 0 } $
$$ \begin{array}{c|rrrrr}8&1&0&\color{orangered}{ -64 }&2&-9\\& & 8& \color{orangered}{64} & & \\ \hline &1&8&\color{orangered}{0}&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 8 } \cdot \color{blue}{ 0 } = \color{blue}{ 0 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{8}&1&0&-64&2&-9\\& & 8& 64& \color{blue}{0} & \\ \hline &1&8&\color{blue}{0}&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 2 } + \color{orangered}{ 0 } = \color{orangered}{ 2 } $
$$ \begin{array}{c|rrrrr}8&1&0&-64&\color{orangered}{ 2 }&-9\\& & 8& 64& \color{orangered}{0} & \\ \hline &1&8&0&\color{orangered}{2}& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 8 } \cdot \color{blue}{ 2 } = \color{blue}{ 16 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{8}&1&0&-64&2&-9\\& & 8& 64& 0& \color{blue}{16} \\ \hline &1&8&0&\color{blue}{2}& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ -9 } + \color{orangered}{ 16 } = \color{orangered}{ 7 } $
$$ \begin{array}{c|rrrrr}8&1&0&-64&2&\color{orangered}{ -9 }\\& & 8& 64& 0& \color{orangered}{16} \\ \hline &\color{blue}{1}&\color{blue}{8}&\color{blue}{0}&\color{blue}{2}&\color{orangered}{7} \end{array} $$Bottom line represents the quotient $ \color{blue}{ x^{3}+8x^{2}+2 } $ with a remainder of $ \color{red}{ 7 } $.