Divide using synthetic division $$ ( 4x^{5}-3x^{3}+x-4 ) \div ( x-6 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrrr}6&4&0&-3&0&1&-4\\& & 24& 144& 846& 5076& \color{black}{30462} \\ \hline &\color{blue}{4}&\color{blue}{24}&\color{blue}{141}&\color{blue}{846}&\color{blue}{5077}&\color{orangered}{30458} \end{array} $$The solution is:
$$ \frac{ 4x^{5}-3x^{3}+x-4 }{ x-6 } = \color{blue}{4x^{4}+24x^{3}+141x^{2}+846x+5077} ~+~ \frac{ \color{red}{ 30458 } }{ x-6 } $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x -6 = 0 $ ( $ x = \color{blue}{ 6 } $ ) at the left.
$$ \begin{array}{c|rrrrrr}\color{blue}{6}&4&0&-3&0&1&-4\\& & & & & & \\ \hline &&&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrrr}6&\color{orangered}{ 4 }&0&-3&0&1&-4\\& & & & & & \\ \hline &\color{orangered}{4}&&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 6 } \cdot \color{blue}{ 4 } = \color{blue}{ 24 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{6}&4&0&-3&0&1&-4\\& & \color{blue}{24} & & & & \\ \hline &\color{blue}{4}&&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 24 } = \color{orangered}{ 24 } $
$$ \begin{array}{c|rrrrrr}6&4&\color{orangered}{ 0 }&-3&0&1&-4\\& & \color{orangered}{24} & & & & \\ \hline &4&\color{orangered}{24}&&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 6 } \cdot \color{blue}{ 24 } = \color{blue}{ 144 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{6}&4&0&-3&0&1&-4\\& & 24& \color{blue}{144} & & & \\ \hline &4&\color{blue}{24}&&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -3 } + \color{orangered}{ 144 } = \color{orangered}{ 141 } $
$$ \begin{array}{c|rrrrrr}6&4&0&\color{orangered}{ -3 }&0&1&-4\\& & 24& \color{orangered}{144} & & & \\ \hline &4&24&\color{orangered}{141}&&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 6 } \cdot \color{blue}{ 141 } = \color{blue}{ 846 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{6}&4&0&-3&0&1&-4\\& & 24& 144& \color{blue}{846} & & \\ \hline &4&24&\color{blue}{141}&&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 846 } = \color{orangered}{ 846 } $
$$ \begin{array}{c|rrrrrr}6&4&0&-3&\color{orangered}{ 0 }&1&-4\\& & 24& 144& \color{orangered}{846} & & \\ \hline &4&24&141&\color{orangered}{846}&& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 6 } \cdot \color{blue}{ 846 } = \color{blue}{ 5076 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{6}&4&0&-3&0&1&-4\\& & 24& 144& 846& \color{blue}{5076} & \\ \hline &4&24&141&\color{blue}{846}&& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ 1 } + \color{orangered}{ 5076 } = \color{orangered}{ 5077 } $
$$ \begin{array}{c|rrrrrr}6&4&0&-3&0&\color{orangered}{ 1 }&-4\\& & 24& 144& 846& \color{orangered}{5076} & \\ \hline &4&24&141&846&\color{orangered}{5077}& \end{array} $$Step 10 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 6 } \cdot \color{blue}{ 5077 } = \color{blue}{ 30462 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{6}&4&0&-3&0&1&-4\\& & 24& 144& 846& 5076& \color{blue}{30462} \\ \hline &4&24&141&846&\color{blue}{5077}& \end{array} $$Step 11 : Add down last column: $ \color{orangered}{ -4 } + \color{orangered}{ 30462 } = \color{orangered}{ 30458 } $
$$ \begin{array}{c|rrrrrr}6&4&0&-3&0&1&\color{orangered}{ -4 }\\& & 24& 144& 846& 5076& \color{orangered}{30462} \\ \hline &\color{blue}{4}&\color{blue}{24}&\color{blue}{141}&\color{blue}{846}&\color{blue}{5077}&\color{orangered}{30458} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 4x^{4}+24x^{3}+141x^{2}+846x+5077 } $ with a remainder of $ \color{red}{ 30458 } $.