Divide using synthetic division $$ ( -4x^{3}+x^{2}+2x+4 ) \div ( x+4 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}-4&-4&1&2&4\\& & 16& -68& \color{black}{264} \\ \hline &\color{blue}{-4}&\color{blue}{17}&\color{blue}{-66}&\color{orangered}{268} \end{array} $$The solution is:
$$ \frac{ -4x^{3}+x^{2}+2x+4 }{ x+4 } = \color{blue}{-4x^{2}+17x-66} ~+~ \frac{ \color{red}{ 268 } }{ x+4 } $$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.
$$ \begin{array}{c|rrrr}\color{blue}{-4}&-4&1&2&4\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}-4&\color{orangered}{ -4 }&1&2&4\\& & & & \\ \hline &\color{orangered}{-4}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -4 } \cdot \color{blue}{ \left( -4 \right) } = \color{blue}{ 16 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-4}&-4&1&2&4\\& & \color{blue}{16} & & \\ \hline &\color{blue}{-4}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 1 } + \color{orangered}{ 16 } = \color{orangered}{ 17 } $
$$ \begin{array}{c|rrrr}-4&-4&\color{orangered}{ 1 }&2&4\\& & \color{orangered}{16} & & \\ \hline &-4&\color{orangered}{17}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -4 } \cdot \color{blue}{ 17 } = \color{blue}{ -68 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-4}&-4&1&2&4\\& & 16& \color{blue}{-68} & \\ \hline &-4&\color{blue}{17}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 2 } + \color{orangered}{ \left( -68 \right) } = \color{orangered}{ -66 } $
$$ \begin{array}{c|rrrr}-4&-4&1&\color{orangered}{ 2 }&4\\& & 16& \color{orangered}{-68} & \\ \hline &-4&17&\color{orangered}{-66}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -4 } \cdot \color{blue}{ \left( -66 \right) } = \color{blue}{ 264 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-4}&-4&1&2&4\\& & 16& -68& \color{blue}{264} \\ \hline &-4&17&\color{blue}{-66}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 4 } + \color{orangered}{ 264 } = \color{orangered}{ 268 } $
$$ \begin{array}{c|rrrr}-4&-4&1&2&\color{orangered}{ 4 }\\& & 16& -68& \color{orangered}{264} \\ \hline &\color{blue}{-4}&\color{blue}{17}&\color{blue}{-66}&\color{orangered}{268} \end{array} $$Bottom line represents the quotient $ \color{blue}{ -4x^{2}+17x-66 } $ with a remainder of $ \color{red}{ 268 } $.