Divide using synthetic division $$ ( 6x^{4}-25x^{2}+4 ) \div ( x-4 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrr}4&6&0&-25&0&4\\& & 24& 96& 284& \color{black}{1136} \\ \hline &\color{blue}{6}&\color{blue}{24}&\color{blue}{71}&\color{blue}{284}&\color{orangered}{1140} \end{array} $$The solution is:
$$ \frac{ 6x^{4}-25x^{2}+4 }{ x-4 } = \color{blue}{6x^{3}+24x^{2}+71x+284} ~+~ \frac{ \color{red}{ 1140 } }{ x-4 } $$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.
$$ \begin{array}{c|rrrrr}\color{blue}{4}&6&0&-25&0&4\\& & & & & \\ \hline &&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrr}4&\color{orangered}{ 6 }&0&-25&0&4\\& & & & & \\ \hline &\color{orangered}{6}&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 4 } \cdot \color{blue}{ 6 } = \color{blue}{ 24 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{4}&6&0&-25&0&4\\& & \color{blue}{24} & & & \\ \hline &\color{blue}{6}&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 24 } = \color{orangered}{ 24 } $
$$ \begin{array}{c|rrrrr}4&6&\color{orangered}{ 0 }&-25&0&4\\& & \color{orangered}{24} & & & \\ \hline &6&\color{orangered}{24}&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 4 } \cdot \color{blue}{ 24 } = \color{blue}{ 96 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{4}&6&0&-25&0&4\\& & 24& \color{blue}{96} & & \\ \hline &6&\color{blue}{24}&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -25 } + \color{orangered}{ 96 } = \color{orangered}{ 71 } $
$$ \begin{array}{c|rrrrr}4&6&0&\color{orangered}{ -25 }&0&4\\& & 24& \color{orangered}{96} & & \\ \hline &6&24&\color{orangered}{71}&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 4 } \cdot \color{blue}{ 71 } = \color{blue}{ 284 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{4}&6&0&-25&0&4\\& & 24& 96& \color{blue}{284} & \\ \hline &6&24&\color{blue}{71}&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 284 } = \color{orangered}{ 284 } $
$$ \begin{array}{c|rrrrr}4&6&0&-25&\color{orangered}{ 0 }&4\\& & 24& 96& \color{orangered}{284} & \\ \hline &6&24&71&\color{orangered}{284}& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 4 } \cdot \color{blue}{ 284 } = \color{blue}{ 1136 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{4}&6&0&-25&0&4\\& & 24& 96& 284& \color{blue}{1136} \\ \hline &6&24&71&\color{blue}{284}& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ 4 } + \color{orangered}{ 1136 } = \color{orangered}{ 1140 } $
$$ \begin{array}{c|rrrrr}4&6&0&-25&0&\color{orangered}{ 4 }\\& & 24& 96& 284& \color{orangered}{1136} \\ \hline &\color{blue}{6}&\color{blue}{24}&\color{blue}{71}&\color{blue}{284}&\color{orangered}{1140} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 6x^{3}+24x^{2}+71x+284 } $ with a remainder of $ \color{red}{ 1140 } $.