Divide using synthetic division $$ ( 6x^{6}-2x^{4}+4x^{2}+8 ) \div ( x-2 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrrrr}2&6&0&-2&0&4&0&8\\& & 12& 24& 44& 88& 184& \color{black}{368} \\ \hline &\color{blue}{6}&\color{blue}{12}&\color{blue}{22}&\color{blue}{44}&\color{blue}{92}&\color{blue}{184}&\color{orangered}{376} \end{array} $$The solution is:
$$ \frac{ 6x^{6}-2x^{4}+4x^{2}+8 }{ x-2 } = \color{blue}{6x^{5}+12x^{4}+22x^{3}+44x^{2}+92x+184} ~+~ \frac{ \color{red}{ 376 } }{ x-2 } $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x -2 = 0 $ ( $ x = \color{blue}{ 2 } $ ) at the left.
$$ \begin{array}{c|rrrrrrr}\color{blue}{2}&6&0&-2&0&4&0&8\\& & & & & & & \\ \hline &&&&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrrrr}2&\color{orangered}{ 6 }&0&-2&0&4&0&8\\& & & & & & & \\ \hline &\color{orangered}{6}&&&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ 6 } = \color{blue}{ 12 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{2}&6&0&-2&0&4&0&8\\& & \color{blue}{12} & & & & & \\ \hline &\color{blue}{6}&&&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 12 } = \color{orangered}{ 12 } $
$$ \begin{array}{c|rrrrrrr}2&6&\color{orangered}{ 0 }&-2&0&4&0&8\\& & \color{orangered}{12} & & & & & \\ \hline &6&\color{orangered}{12}&&&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ 12 } = \color{blue}{ 24 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{2}&6&0&-2&0&4&0&8\\& & 12& \color{blue}{24} & & & & \\ \hline &6&\color{blue}{12}&&&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -2 } + \color{orangered}{ 24 } = \color{orangered}{ 22 } $
$$ \begin{array}{c|rrrrrrr}2&6&0&\color{orangered}{ -2 }&0&4&0&8\\& & 12& \color{orangered}{24} & & & & \\ \hline &6&12&\color{orangered}{22}&&&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ 22 } = \color{blue}{ 44 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{2}&6&0&-2&0&4&0&8\\& & 12& 24& \color{blue}{44} & & & \\ \hline &6&12&\color{blue}{22}&&&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 44 } = \color{orangered}{ 44 } $
$$ \begin{array}{c|rrrrrrr}2&6&0&-2&\color{orangered}{ 0 }&4&0&8\\& & 12& 24& \color{orangered}{44} & & & \\ \hline &6&12&22&\color{orangered}{44}&&& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ 44 } = \color{blue}{ 88 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{2}&6&0&-2&0&4&0&8\\& & 12& 24& 44& \color{blue}{88} & & \\ \hline &6&12&22&\color{blue}{44}&&& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ 4 } + \color{orangered}{ 88 } = \color{orangered}{ 92 } $
$$ \begin{array}{c|rrrrrrr}2&6&0&-2&0&\color{orangered}{ 4 }&0&8\\& & 12& 24& 44& \color{orangered}{88} & & \\ \hline &6&12&22&44&\color{orangered}{92}&& \end{array} $$Step 10 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ 92 } = \color{blue}{ 184 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{2}&6&0&-2&0&4&0&8\\& & 12& 24& 44& 88& \color{blue}{184} & \\ \hline &6&12&22&44&\color{blue}{92}&& \end{array} $$Step 11 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 184 } = \color{orangered}{ 184 } $
$$ \begin{array}{c|rrrrrrr}2&6&0&-2&0&4&\color{orangered}{ 0 }&8\\& & 12& 24& 44& 88& \color{orangered}{184} & \\ \hline &6&12&22&44&92&\color{orangered}{184}& \end{array} $$Step 12 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ 184 } = \color{blue}{ 368 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{2}&6&0&-2&0&4&0&8\\& & 12& 24& 44& 88& 184& \color{blue}{368} \\ \hline &6&12&22&44&92&\color{blue}{184}& \end{array} $$Step 13 : Add down last column: $ \color{orangered}{ 8 } + \color{orangered}{ 368 } = \color{orangered}{ 376 } $
$$ \begin{array}{c|rrrrrrr}2&6&0&-2&0&4&0&\color{orangered}{ 8 }\\& & 12& 24& 44& 88& 184& \color{orangered}{368} \\ \hline &\color{blue}{6}&\color{blue}{12}&\color{blue}{22}&\color{blue}{44}&\color{blue}{92}&\color{blue}{184}&\color{orangered}{376} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 6x^{5}+12x^{4}+22x^{3}+44x^{2}+92x+184 } $ with a remainder of $ \color{red}{ 376 } $.