Divide using synthetic division $$ ( -64x^{4}+x^{2}-15 ) \div ( x+4 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrr}-4&-64&0&1&0&-15\\& & 256& -1024& 4092& \color{black}{-16368} \\ \hline &\color{blue}{-64}&\color{blue}{256}&\color{blue}{-1023}&\color{blue}{4092}&\color{orangered}{-16383} \end{array} $$The solution is:
$$ \frac{ -64x^{4}+x^{2}-15 }{ x+4 } = \color{blue}{-64x^{3}+256x^{2}-1023x+4092} \color{red}{~-~} \frac{ \color{red}{ 16383 } }{ x+4 } $$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.
$$ \begin{array}{c|rrrrr}\color{blue}{-4}&-64&0&1&0&-15\\& & & & & \\ \hline &&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrr}-4&\color{orangered}{ -64 }&0&1&0&-15\\& & & & & \\ \hline &\color{orangered}{-64}&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -4 } \cdot \color{blue}{ \left( -64 \right) } = \color{blue}{ 256 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-4}&-64&0&1&0&-15\\& & \color{blue}{256} & & & \\ \hline &\color{blue}{-64}&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 256 } = \color{orangered}{ 256 } $
$$ \begin{array}{c|rrrrr}-4&-64&\color{orangered}{ 0 }&1&0&-15\\& & \color{orangered}{256} & & & \\ \hline &-64&\color{orangered}{256}&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -4 } \cdot \color{blue}{ 256 } = \color{blue}{ -1024 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-4}&-64&0&1&0&-15\\& & 256& \color{blue}{-1024} & & \\ \hline &-64&\color{blue}{256}&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 1 } + \color{orangered}{ \left( -1024 \right) } = \color{orangered}{ -1023 } $
$$ \begin{array}{c|rrrrr}-4&-64&0&\color{orangered}{ 1 }&0&-15\\& & 256& \color{orangered}{-1024} & & \\ \hline &-64&256&\color{orangered}{-1023}&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -4 } \cdot \color{blue}{ \left( -1023 \right) } = \color{blue}{ 4092 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-4}&-64&0&1&0&-15\\& & 256& -1024& \color{blue}{4092} & \\ \hline &-64&256&\color{blue}{-1023}&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 4092 } = \color{orangered}{ 4092 } $
$$ \begin{array}{c|rrrrr}-4&-64&0&1&\color{orangered}{ 0 }&-15\\& & 256& -1024& \color{orangered}{4092} & \\ \hline &-64&256&-1023&\color{orangered}{4092}& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -4 } \cdot \color{blue}{ 4092 } = \color{blue}{ -16368 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-4}&-64&0&1&0&-15\\& & 256& -1024& 4092& \color{blue}{-16368} \\ \hline &-64&256&-1023&\color{blue}{4092}& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ -15 } + \color{orangered}{ \left( -16368 \right) } = \color{orangered}{ -16383 } $
$$ \begin{array}{c|rrrrr}-4&-64&0&1&0&\color{orangered}{ -15 }\\& & 256& -1024& 4092& \color{orangered}{-16368} \\ \hline &\color{blue}{-64}&\color{blue}{256}&\color{blue}{-1023}&\color{blue}{4092}&\color{orangered}{-16383} \end{array} $$Bottom line represents the quotient $ \color{blue}{ -64x^{3}+256x^{2}-1023x+4092 } $ with a remainder of $ \color{red}{ -16383 } $.