Divide using synthetic division $$ ( 2x^{6}-64x^{4}+x^{2}-15 ) \div ( x+4 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrrrr}-4&2&0&-64&0&1&0&-15\\& & -8& 32& 128& -512& 2044& \color{black}{-8176} \\ \hline &\color{blue}{2}&\color{blue}{-8}&\color{blue}{-32}&\color{blue}{128}&\color{blue}{-511}&\color{blue}{2044}&\color{orangered}{-8191} \end{array} $$The solution is:
$$ \frac{ 2x^{6}-64x^{4}+x^{2}-15 }{ x+4 } = \color{blue}{2x^{5}-8x^{4}-32x^{3}+128x^{2}-511x+2044} \color{red}{~-~} \frac{ \color{red}{ 8191 } }{ x+4 } $$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.
$$ \begin{array}{c|rrrrrrr}\color{blue}{-4}&2&0&-64&0&1&0&-15\\& & & & & & & \\ \hline &&&&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrrrr}-4&\color{orangered}{ 2 }&0&-64&0&1&0&-15\\& & & & & & & \\ \hline &\color{orangered}{2}&&&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -4 } \cdot \color{blue}{ 2 } = \color{blue}{ -8 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{-4}&2&0&-64&0&1&0&-15\\& & \color{blue}{-8} & & & & & \\ \hline &\color{blue}{2}&&&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ \left( -8 \right) } = \color{orangered}{ -8 } $
$$ \begin{array}{c|rrrrrrr}-4&2&\color{orangered}{ 0 }&-64&0&1&0&-15\\& & \color{orangered}{-8} & & & & & \\ \hline &2&\color{orangered}{-8}&&&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -4 } \cdot \color{blue}{ \left( -8 \right) } = \color{blue}{ 32 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{-4}&2&0&-64&0&1&0&-15\\& & -8& \color{blue}{32} & & & & \\ \hline &2&\color{blue}{-8}&&&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -64 } + \color{orangered}{ 32 } = \color{orangered}{ -32 } $
$$ \begin{array}{c|rrrrrrr}-4&2&0&\color{orangered}{ -64 }&0&1&0&-15\\& & -8& \color{orangered}{32} & & & & \\ \hline &2&-8&\color{orangered}{-32}&&&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -4 } \cdot \color{blue}{ \left( -32 \right) } = \color{blue}{ 128 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{-4}&2&0&-64&0&1&0&-15\\& & -8& 32& \color{blue}{128} & & & \\ \hline &2&-8&\color{blue}{-32}&&&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 128 } = \color{orangered}{ 128 } $
$$ \begin{array}{c|rrrrrrr}-4&2&0&-64&\color{orangered}{ 0 }&1&0&-15\\& & -8& 32& \color{orangered}{128} & & & \\ \hline &2&-8&-32&\color{orangered}{128}&&& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -4 } \cdot \color{blue}{ 128 } = \color{blue}{ -512 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{-4}&2&0&-64&0&1&0&-15\\& & -8& 32& 128& \color{blue}{-512} & & \\ \hline &2&-8&-32&\color{blue}{128}&&& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ 1 } + \color{orangered}{ \left( -512 \right) } = \color{orangered}{ -511 } $
$$ \begin{array}{c|rrrrrrr}-4&2&0&-64&0&\color{orangered}{ 1 }&0&-15\\& & -8& 32& 128& \color{orangered}{-512} & & \\ \hline &2&-8&-32&128&\color{orangered}{-511}&& \end{array} $$Step 10 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -4 } \cdot \color{blue}{ \left( -511 \right) } = \color{blue}{ 2044 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{-4}&2&0&-64&0&1&0&-15\\& & -8& 32& 128& -512& \color{blue}{2044} & \\ \hline &2&-8&-32&128&\color{blue}{-511}&& \end{array} $$Step 11 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 2044 } = \color{orangered}{ 2044 } $
$$ \begin{array}{c|rrrrrrr}-4&2&0&-64&0&1&\color{orangered}{ 0 }&-15\\& & -8& 32& 128& -512& \color{orangered}{2044} & \\ \hline &2&-8&-32&128&-511&\color{orangered}{2044}& \end{array} $$Step 12 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -4 } \cdot \color{blue}{ 2044 } = \color{blue}{ -8176 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{-4}&2&0&-64&0&1&0&-15\\& & -8& 32& 128& -512& 2044& \color{blue}{-8176} \\ \hline &2&-8&-32&128&-511&\color{blue}{2044}& \end{array} $$Step 13 : Add down last column: $ \color{orangered}{ -15 } + \color{orangered}{ \left( -8176 \right) } = \color{orangered}{ -8191 } $
$$ \begin{array}{c|rrrrrrr}-4&2&0&-64&0&1&0&\color{orangered}{ -15 }\\& & -8& 32& 128& -512& 2044& \color{orangered}{-8176} \\ \hline &\color{blue}{2}&\color{blue}{-8}&\color{blue}{-32}&\color{blue}{128}&\color{blue}{-511}&\color{blue}{2044}&\color{orangered}{-8191} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 2x^{5}-8x^{4}-32x^{3}+128x^{2}-511x+2044 } $ with a remainder of $ \color{red}{ -8191 } $.