Divide using synthetic division $$ ( 4x^{5}-10x^{4}+8x^{3}+x^{2}-8 ) \div ( x-8 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrrr}8&4&-10&8&1&0&-8\\& & 32& 176& 1472& 11784& \color{black}{94272} \\ \hline &\color{blue}{4}&\color{blue}{22}&\color{blue}{184}&\color{blue}{1473}&\color{blue}{11784}&\color{orangered}{94264} \end{array} $$The solution is:
$$ \frac{ 4x^{5}-10x^{4}+8x^{3}+x^{2}-8 }{ x-8 } = \color{blue}{4x^{4}+22x^{3}+184x^{2}+1473x+11784} ~+~ \frac{ \color{red}{ 94264 } }{ x-8 } $$Explanation
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.
$$ \begin{array}{c|rrrrrr}\color{blue}{8}&4&-10&8&1&0&-8\\& & & & & & \\ \hline &&&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrrr}8&\color{orangered}{ 4 }&-10&8&1&0&-8\\& & & & & & \\ \hline &\color{orangered}{4}&&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 8 } \cdot \color{blue}{ 4 } = \color{blue}{ 32 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{8}&4&-10&8&1&0&-8\\& & \color{blue}{32} & & & & \\ \hline &\color{blue}{4}&&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -10 } + \color{orangered}{ 32 } = \color{orangered}{ 22 } $
$$ \begin{array}{c|rrrrrr}8&4&\color{orangered}{ -10 }&8&1&0&-8\\& & \color{orangered}{32} & & & & \\ \hline &4&\color{orangered}{22}&&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 8 } \cdot \color{blue}{ 22 } = \color{blue}{ 176 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{8}&4&-10&8&1&0&-8\\& & 32& \color{blue}{176} & & & \\ \hline &4&\color{blue}{22}&&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 8 } + \color{orangered}{ 176 } = \color{orangered}{ 184 } $
$$ \begin{array}{c|rrrrrr}8&4&-10&\color{orangered}{ 8 }&1&0&-8\\& & 32& \color{orangered}{176} & & & \\ \hline &4&22&\color{orangered}{184}&&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 8 } \cdot \color{blue}{ 184 } = \color{blue}{ 1472 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{8}&4&-10&8&1&0&-8\\& & 32& 176& \color{blue}{1472} & & \\ \hline &4&22&\color{blue}{184}&&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 1 } + \color{orangered}{ 1472 } = \color{orangered}{ 1473 } $
$$ \begin{array}{c|rrrrrr}8&4&-10&8&\color{orangered}{ 1 }&0&-8\\& & 32& 176& \color{orangered}{1472} & & \\ \hline &4&22&184&\color{orangered}{1473}&& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 8 } \cdot \color{blue}{ 1473 } = \color{blue}{ 11784 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{8}&4&-10&8&1&0&-8\\& & 32& 176& 1472& \color{blue}{11784} & \\ \hline &4&22&184&\color{blue}{1473}&& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 11784 } = \color{orangered}{ 11784 } $
$$ \begin{array}{c|rrrrrr}8&4&-10&8&1&\color{orangered}{ 0 }&-8\\& & 32& 176& 1472& \color{orangered}{11784} & \\ \hline &4&22&184&1473&\color{orangered}{11784}& \end{array} $$Step 10 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 8 } \cdot \color{blue}{ 11784 } = \color{blue}{ 94272 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{8}&4&-10&8&1&0&-8\\& & 32& 176& 1472& 11784& \color{blue}{94272} \\ \hline &4&22&184&1473&\color{blue}{11784}& \end{array} $$Step 11 : Add down last column: $ \color{orangered}{ -8 } + \color{orangered}{ 94272 } = \color{orangered}{ 94264 } $
$$ \begin{array}{c|rrrrrr}8&4&-10&8&1&0&\color{orangered}{ -8 }\\& & 32& 176& 1472& 11784& \color{orangered}{94272} \\ \hline &\color{blue}{4}&\color{blue}{22}&\color{blue}{184}&\color{blue}{1473}&\color{blue}{11784}&\color{orangered}{94264} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 4x^{4}+22x^{3}+184x^{2}+1473x+11784 } $ with a remainder of $ \color{red}{ 94264 } $.