Check if $ x+4 $ is a factor of $ 4x^{6}-64x^{4}+3x^{2}-48 $.
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrrrr}-4&4&0&-64&0&3&0&-48\\& & -16& 64& 0& 0& -12& \color{black}{48} \\ \hline &\color{blue}{4}&\color{blue}{-16}&\color{blue}{0}&\color{blue}{0}&\color{blue}{3}&\color{blue}{-12}&\color{orangered}{0} \end{array} $$Because the remainder equals zero, we conclude that the $ x+4 $ is a factor of the $ 4x^{6}-64x^{4}+3x^{2}-48 $.
Explanation
First we need to create a synthetic division table.
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.
$$ \begin{array}{c|rrrrrrr}\color{blue}{-4}&4&0&-64&0&3&0&-48\\& & & & & & & \\ \hline &&&&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrrrr}-4&\color{orangered}{ 4 }&0&-64&0&3&0&-48\\& & & & & & & \\ \hline &\color{orangered}{4}&&&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -4 } \cdot \color{blue}{ 4 } = \color{blue}{ -16 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{-4}&4&0&-64&0&3&0&-48\\& & \color{blue}{-16} & & & & & \\ \hline &\color{blue}{4}&&&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ \left( -16 \right) } = \color{orangered}{ -16 } $
$$ \begin{array}{c|rrrrrrr}-4&4&\color{orangered}{ 0 }&-64&0&3&0&-48\\& & \color{orangered}{-16} & & & & & \\ \hline &4&\color{orangered}{-16}&&&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -4 } \cdot \color{blue}{ \left( -16 \right) } = \color{blue}{ 64 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{-4}&4&0&-64&0&3&0&-48\\& & -16& \color{blue}{64} & & & & \\ \hline &4&\color{blue}{-16}&&&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -64 } + \color{orangered}{ 64 } = \color{orangered}{ 0 } $
$$ \begin{array}{c|rrrrrrr}-4&4&0&\color{orangered}{ -64 }&0&3&0&-48\\& & -16& \color{orangered}{64} & & & & \\ \hline &4&-16&\color{orangered}{0}&&&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -4 } \cdot \color{blue}{ 0 } = \color{blue}{ 0 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{-4}&4&0&-64&0&3&0&-48\\& & -16& 64& \color{blue}{0} & & & \\ \hline &4&-16&\color{blue}{0}&&&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 0 } = \color{orangered}{ 0 } $
$$ \begin{array}{c|rrrrrrr}-4&4&0&-64&\color{orangered}{ 0 }&3&0&-48\\& & -16& 64& \color{orangered}{0} & & & \\ \hline &4&-16&0&\color{orangered}{0}&&& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -4 } \cdot \color{blue}{ 0 } = \color{blue}{ 0 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{-4}&4&0&-64&0&3&0&-48\\& & -16& 64& 0& \color{blue}{0} & & \\ \hline &4&-16&0&\color{blue}{0}&&& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ 3 } + \color{orangered}{ 0 } = \color{orangered}{ 3 } $
$$ \begin{array}{c|rrrrrrr}-4&4&0&-64&0&\color{orangered}{ 3 }&0&-48\\& & -16& 64& 0& \color{orangered}{0} & & \\ \hline &4&-16&0&0&\color{orangered}{3}&& \end{array} $$Step 10 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -4 } \cdot \color{blue}{ 3 } = \color{blue}{ -12 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{-4}&4&0&-64&0&3&0&-48\\& & -16& 64& 0& 0& \color{blue}{-12} & \\ \hline &4&-16&0&0&\color{blue}{3}&& \end{array} $$Step 11 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ \left( -12 \right) } = \color{orangered}{ -12 } $
$$ \begin{array}{c|rrrrrrr}-4&4&0&-64&0&3&\color{orangered}{ 0 }&-48\\& & -16& 64& 0& 0& \color{orangered}{-12} & \\ \hline &4&-16&0&0&3&\color{orangered}{-12}& \end{array} $$Step 12 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -4 } \cdot \color{blue}{ \left( -12 \right) } = \color{blue}{ 48 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{-4}&4&0&-64&0&3&0&-48\\& & -16& 64& 0& 0& -12& \color{blue}{48} \\ \hline &4&-16&0&0&3&\color{blue}{-12}& \end{array} $$Step 13 : Add down last column: $ \color{orangered}{ -48 } + \color{orangered}{ 48 } = \color{orangered}{ 0 } $
$$ \begin{array}{c|rrrrrrr}-4&4&0&-64&0&3&0&\color{orangered}{ -48 }\\& & -16& 64& 0& 0& -12& \color{orangered}{48} \\ \hline &\color{blue}{4}&\color{blue}{-16}&\color{blue}{0}&\color{blue}{0}&\color{blue}{3}&\color{blue}{-12}&\color{orangered}{0} \end{array} $$Remainder is the last entry in the bottom row $ \left(\color{red}{ 0 }\right)$.