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