Divide using synthetic division $$ ( -x^{4}+16 ) \div ( x+4 ) $$
The synthetic division table is:
$$ \begin{array}{c|rrrrr}-4&-1&0&0&0&16\\& & 4& -16& 64& \color{black}{-256} \\ \hline &\color{blue}{-1}&\color{blue}{4}&\color{blue}{-16}&\color{blue}{64}&\color{orangered}{-240} \end{array} $$The solution is:
$$ \dfrac{ -x^{4}+16 }{ x+4 } = \color{blue}{-x^{3}+4x^{2}-16x+64} \color{red}{~-~} \dfrac{ \color{red}{ 240 } }{ x+4 } $$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.
$$ \begin{array}{c|rrrrr}\color{blue}{-4}&-1&0&0&0&16\\& & & & & \\ \hline &&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrr}-4&\color{orangered}{ -1 }&0&0&0&16\\& & & & & \\ \hline &\color{orangered}{-1}&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -4 } \cdot \color{blue}{ \left( -1 \right) } = \color{blue}{ 4 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-4}&-1&0&0&0&16\\& & \color{blue}{4} & & & \\ \hline &\color{blue}{-1}&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 4 } = \color{orangered}{ 4 } $
$$ \begin{array}{c|rrrrr}-4&-1&\color{orangered}{ 0 }&0&0&16\\& & \color{orangered}{4} & & & \\ \hline &-1&\color{orangered}{4}&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -4 } \cdot \color{blue}{ 4 } = \color{blue}{ -16 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-4}&-1&0&0&0&16\\& & 4& \color{blue}{-16} & & \\ \hline &-1&\color{blue}{4}&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ \left( -16 \right) } = \color{orangered}{ -16 } $
$$ \begin{array}{c|rrrrr}-4&-1&0&\color{orangered}{ 0 }&0&16\\& & 4& \color{orangered}{-16} & & \\ \hline &-1&4&\color{orangered}{-16}&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -4 } \cdot \color{blue}{ \left( -16 \right) } = \color{blue}{ 64 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-4}&-1&0&0&0&16\\& & 4& -16& \color{blue}{64} & \\ \hline &-1&4&\color{blue}{-16}&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 64 } = \color{orangered}{ 64 } $
$$ \begin{array}{c|rrrrr}-4&-1&0&0&\color{orangered}{ 0 }&16\\& & 4& -16& \color{orangered}{64} & \\ \hline &-1&4&-16&\color{orangered}{64}& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -4 } \cdot \color{blue}{ 64 } = \color{blue}{ -256 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-4}&-1&0&0&0&16\\& & 4& -16& 64& \color{blue}{-256} \\ \hline &-1&4&-16&\color{blue}{64}& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ 16 } + \color{orangered}{ \left( -256 \right) } = \color{orangered}{ -240 } $
$$ \begin{array}{c|rrrrr}-4&-1&0&0&0&\color{orangered}{ 16 }\\& & 4& -16& 64& \color{orangered}{-256} \\ \hline &\color{blue}{-1}&\color{blue}{4}&\color{blue}{-16}&\color{blue}{64}&\color{orangered}{-240} \end{array} $$Bottom line represents the quotient $ \color{blue}{ -x^{3}+4x^{2}-16x+64 } $ with a remainder of $ \color{red}{ -240 } $.