Divide using synthetic division $$ ( -4x^{4}+x^{3}-x^{2}+2 ) \div ( x+4 ) $$
The synthetic division table is:
$$ \begin{array}{c|rrrrr}-4&-4&1&-1&0&2\\& & 16& -68& 276& \color{black}{-1104} \\ \hline &\color{blue}{-4}&\color{blue}{17}&\color{blue}{-69}&\color{blue}{276}&\color{orangered}{-1102} \end{array} $$The solution is:
$$ \dfrac{ -4x^{4}+x^{3}-x^{2}+2 }{ x+4 } = \color{blue}{-4x^{3}+17x^{2}-69x+276} \color{red}{~-~} \dfrac{ \color{red}{ 1102 } }{ x+4 } $$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.
$$ \begin{array}{c|rrrrr}\color{blue}{-4}&-4&1&-1&0&2\\& & & & & \\ \hline &&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrr}-4&\color{orangered}{ -4 }&1&-1&0&2\\& & & & & \\ \hline &\color{orangered}{-4}&&&& \end{array} $$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 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-4}&-4&1&-1&0&2\\& & \color{blue}{16} & & & \\ \hline &\color{blue}{-4}&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 1 } + \color{orangered}{ 16 } = \color{orangered}{ 17 } $
$$ \begin{array}{c|rrrrr}-4&-4&\color{orangered}{ 1 }&-1&0&2\\& & \color{orangered}{16} & & & \\ \hline &-4&\color{orangered}{17}&&& \end{array} $$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 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-4}&-4&1&-1&0&2\\& & 16& \color{blue}{-68} & & \\ \hline &-4&\color{blue}{17}&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -1 } + \color{orangered}{ \left( -68 \right) } = \color{orangered}{ -69 } $
$$ \begin{array}{c|rrrrr}-4&-4&1&\color{orangered}{ -1 }&0&2\\& & 16& \color{orangered}{-68} & & \\ \hline &-4&17&\color{orangered}{-69}&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -4 } \cdot \color{blue}{ \left( -69 \right) } = \color{blue}{ 276 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-4}&-4&1&-1&0&2\\& & 16& -68& \color{blue}{276} & \\ \hline &-4&17&\color{blue}{-69}&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 276 } = \color{orangered}{ 276 } $
$$ \begin{array}{c|rrrrr}-4&-4&1&-1&\color{orangered}{ 0 }&2\\& & 16& -68& \color{orangered}{276} & \\ \hline &-4&17&-69&\color{orangered}{276}& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -4 } \cdot \color{blue}{ 276 } = \color{blue}{ -1104 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-4}&-4&1&-1&0&2\\& & 16& -68& 276& \color{blue}{-1104} \\ \hline &-4&17&-69&\color{blue}{276}& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ 2 } + \color{orangered}{ \left( -1104 \right) } = \color{orangered}{ -1102 } $
$$ \begin{array}{c|rrrrr}-4&-4&1&-1&0&\color{orangered}{ 2 }\\& & 16& -68& 276& \color{orangered}{-1104} \\ \hline &\color{blue}{-4}&\color{blue}{17}&\color{blue}{-69}&\color{blue}{276}&\color{orangered}{-1102} \end{array} $$Bottom line represents the quotient $ \color{blue}{ -4x^{3}+17x^{2}-69x+276 } $ with a remainder of $ \color{red}{ -1102 } $.