Divide using synthetic division $$ ( 4x^{5}-10x^{4}+8x^{3}+x^{2}-8 ) \div ( x-4 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrrr}4&4&-10&8&1&0&-8\\& & 16& 24& 128& 516& \color{black}{2064} \\ \hline &\color{blue}{4}&\color{blue}{6}&\color{blue}{32}&\color{blue}{129}&\color{blue}{516}&\color{orangered}{2056} \end{array} $$The solution is:
$$ \frac{ 4x^{5}-10x^{4}+8x^{3}+x^{2}-8 }{ x-4 } = \color{blue}{4x^{4}+6x^{3}+32x^{2}+129x+516} ~+~ \frac{ \color{red}{ 2056 } }{ x-4 } $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x -4 = 0 $ ( $ x = \color{blue}{ 4 } $ ) at the left.
$$ \begin{array}{c|rrrrrr}\color{blue}{4}&4&-10&8&1&0&-8\\& & & & & & \\ \hline &&&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrrr}4&\color{orangered}{ 4 }&-10&8&1&0&-8\\& & & & & & \\ \hline &\color{orangered}{4}&&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 4 } \cdot \color{blue}{ 4 } = \color{blue}{ 16 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{4}&4&-10&8&1&0&-8\\& & \color{blue}{16} & & & & \\ \hline &\color{blue}{4}&&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -10 } + \color{orangered}{ 16 } = \color{orangered}{ 6 } $
$$ \begin{array}{c|rrrrrr}4&4&\color{orangered}{ -10 }&8&1&0&-8\\& & \color{orangered}{16} & & & & \\ \hline &4&\color{orangered}{6}&&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 4 } \cdot \color{blue}{ 6 } = \color{blue}{ 24 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{4}&4&-10&8&1&0&-8\\& & 16& \color{blue}{24} & & & \\ \hline &4&\color{blue}{6}&&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 8 } + \color{orangered}{ 24 } = \color{orangered}{ 32 } $
$$ \begin{array}{c|rrrrrr}4&4&-10&\color{orangered}{ 8 }&1&0&-8\\& & 16& \color{orangered}{24} & & & \\ \hline &4&6&\color{orangered}{32}&&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 4 } \cdot \color{blue}{ 32 } = \color{blue}{ 128 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{4}&4&-10&8&1&0&-8\\& & 16& 24& \color{blue}{128} & & \\ \hline &4&6&\color{blue}{32}&&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 1 } + \color{orangered}{ 128 } = \color{orangered}{ 129 } $
$$ \begin{array}{c|rrrrrr}4&4&-10&8&\color{orangered}{ 1 }&0&-8\\& & 16& 24& \color{orangered}{128} & & \\ \hline &4&6&32&\color{orangered}{129}&& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 4 } \cdot \color{blue}{ 129 } = \color{blue}{ 516 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{4}&4&-10&8&1&0&-8\\& & 16& 24& 128& \color{blue}{516} & \\ \hline &4&6&32&\color{blue}{129}&& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 516 } = \color{orangered}{ 516 } $
$$ \begin{array}{c|rrrrrr}4&4&-10&8&1&\color{orangered}{ 0 }&-8\\& & 16& 24& 128& \color{orangered}{516} & \\ \hline &4&6&32&129&\color{orangered}{516}& \end{array} $$Step 10 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 4 } \cdot \color{blue}{ 516 } = \color{blue}{ 2064 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{4}&4&-10&8&1&0&-8\\& & 16& 24& 128& 516& \color{blue}{2064} \\ \hline &4&6&32&129&\color{blue}{516}& \end{array} $$Step 11 : Add down last column: $ \color{orangered}{ -8 } + \color{orangered}{ 2064 } = \color{orangered}{ 2056 } $
$$ \begin{array}{c|rrrrrr}4&4&-10&8&1&0&\color{orangered}{ -8 }\\& & 16& 24& 128& 516& \color{orangered}{2064} \\ \hline &\color{blue}{4}&\color{blue}{6}&\color{blue}{32}&\color{blue}{129}&\color{blue}{516}&\color{orangered}{2056} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 4x^{4}+6x^{3}+32x^{2}+129x+516 } $ with a remainder of $ \color{red}{ 2056 } $.