Divide using synthetic division $$ ( x^{6}-13x^{4}-52x^{2}+64 ) \div ( x+2 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrrrr}-2&1&0&-13&0&-52&0&64\\& & -2& 4& 18& -36& 176& \color{black}{-352} \\ \hline &\color{blue}{1}&\color{blue}{-2}&\color{blue}{-9}&\color{blue}{18}&\color{blue}{-88}&\color{blue}{176}&\color{orangered}{-288} \end{array} $$The solution is:
$$ \frac{ x^{6}-13x^{4}-52x^{2}+64 }{ x+2 } = \color{blue}{x^{5}-2x^{4}-9x^{3}+18x^{2}-88x+176} \color{red}{~-~} \frac{ \color{red}{ 288 } }{ x+2 } $$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.
$$ \begin{array}{c|rrrrrrr}\color{blue}{-2}&1&0&-13&0&-52&0&64\\& & & & & & & \\ \hline &&&&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrrrr}-2&\color{orangered}{ 1 }&0&-13&0&-52&0&64\\& & & & & & & \\ \hline &\color{orangered}{1}&&&&&& \end{array} $$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 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{-2}&1&0&-13&0&-52&0&64\\& & \color{blue}{-2} & & & & & \\ \hline &\color{blue}{1}&&&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ \left( -2 \right) } = \color{orangered}{ -2 } $
$$ \begin{array}{c|rrrrrrr}-2&1&\color{orangered}{ 0 }&-13&0&-52&0&64\\& & \color{orangered}{-2} & & & & & \\ \hline &1&\color{orangered}{-2}&&&&& \end{array} $$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 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{-2}&1&0&-13&0&-52&0&64\\& & -2& \color{blue}{4} & & & & \\ \hline &1&\color{blue}{-2}&&&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -13 } + \color{orangered}{ 4 } = \color{orangered}{ -9 } $
$$ \begin{array}{c|rrrrrrr}-2&1&0&\color{orangered}{ -13 }&0&-52&0&64\\& & -2& \color{orangered}{4} & & & & \\ \hline &1&-2&\color{orangered}{-9}&&&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ \left( -9 \right) } = \color{blue}{ 18 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{-2}&1&0&-13&0&-52&0&64\\& & -2& 4& \color{blue}{18} & & & \\ \hline &1&-2&\color{blue}{-9}&&&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 18 } = \color{orangered}{ 18 } $
$$ \begin{array}{c|rrrrrrr}-2&1&0&-13&\color{orangered}{ 0 }&-52&0&64\\& & -2& 4& \color{orangered}{18} & & & \\ \hline &1&-2&-9&\color{orangered}{18}&&& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ 18 } = \color{blue}{ -36 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{-2}&1&0&-13&0&-52&0&64\\& & -2& 4& 18& \color{blue}{-36} & & \\ \hline &1&-2&-9&\color{blue}{18}&&& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ -52 } + \color{orangered}{ \left( -36 \right) } = \color{orangered}{ -88 } $
$$ \begin{array}{c|rrrrrrr}-2&1&0&-13&0&\color{orangered}{ -52 }&0&64\\& & -2& 4& 18& \color{orangered}{-36} & & \\ \hline &1&-2&-9&18&\color{orangered}{-88}&& \end{array} $$Step 10 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ \left( -88 \right) } = \color{blue}{ 176 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{-2}&1&0&-13&0&-52&0&64\\& & -2& 4& 18& -36& \color{blue}{176} & \\ \hline &1&-2&-9&18&\color{blue}{-88}&& \end{array} $$Step 11 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 176 } = \color{orangered}{ 176 } $
$$ \begin{array}{c|rrrrrrr}-2&1&0&-13&0&-52&\color{orangered}{ 0 }&64\\& & -2& 4& 18& -36& \color{orangered}{176} & \\ \hline &1&-2&-9&18&-88&\color{orangered}{176}& \end{array} $$Step 12 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ 176 } = \color{blue}{ -352 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{-2}&1&0&-13&0&-52&0&64\\& & -2& 4& 18& -36& 176& \color{blue}{-352} \\ \hline &1&-2&-9&18&-88&\color{blue}{176}& \end{array} $$Step 13 : Add down last column: $ \color{orangered}{ 64 } + \color{orangered}{ \left( -352 \right) } = \color{orangered}{ -288 } $
$$ \begin{array}{c|rrrrrrr}-2&1&0&-13&0&-52&0&\color{orangered}{ 64 }\\& & -2& 4& 18& -36& 176& \color{orangered}{-352} \\ \hline &\color{blue}{1}&\color{blue}{-2}&\color{blue}{-9}&\color{blue}{18}&\color{blue}{-88}&\color{blue}{176}&\color{orangered}{-288} \end{array} $$Bottom line represents the quotient $ \color{blue}{ x^{5}-2x^{4}-9x^{3}+18x^{2}-88x+176 } $ with a remainder of $ \color{red}{ -288 } $.