Divide using synthetic division $$ ( -2x^{4}+11x^{3}+8x^{2}-12x ) \div ( x+4 ) $$
The synthetic division table is:
$$ \begin{array}{c|rrrrr}-4&-2&11&8&-12&0\\& & 8& -76& 272& \color{black}{-1040} \\ \hline &\color{blue}{-2}&\color{blue}{19}&\color{blue}{-68}&\color{blue}{260}&\color{orangered}{-1040} \end{array} $$The solution is:
$$ \dfrac{ -2x^{4}+11x^{3}+8x^{2}-12x }{ x+4 } = \color{blue}{-2x^{3}+19x^{2}-68x+260} \color{red}{~-~} \dfrac{ \color{red}{ 1040 } }{ x+4 } $$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.
$$ \begin{array}{c|rrrrr}\color{blue}{-4}&-2&11&8&-12&0\\& & & & & \\ \hline &&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrr}-4&\color{orangered}{ -2 }&11&8&-12&0\\& & & & & \\ \hline &\color{orangered}{-2}&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -4 } \cdot \color{blue}{ \left( -2 \right) } = \color{blue}{ 8 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-4}&-2&11&8&-12&0\\& & \color{blue}{8} & & & \\ \hline &\color{blue}{-2}&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 11 } + \color{orangered}{ 8 } = \color{orangered}{ 19 } $
$$ \begin{array}{c|rrrrr}-4&-2&\color{orangered}{ 11 }&8&-12&0\\& & \color{orangered}{8} & & & \\ \hline &-2&\color{orangered}{19}&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -4 } \cdot \color{blue}{ 19 } = \color{blue}{ -76 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-4}&-2&11&8&-12&0\\& & 8& \color{blue}{-76} & & \\ \hline &-2&\color{blue}{19}&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 8 } + \color{orangered}{ \left( -76 \right) } = \color{orangered}{ -68 } $
$$ \begin{array}{c|rrrrr}-4&-2&11&\color{orangered}{ 8 }&-12&0\\& & 8& \color{orangered}{-76} & & \\ \hline &-2&19&\color{orangered}{-68}&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -4 } \cdot \color{blue}{ \left( -68 \right) } = \color{blue}{ 272 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-4}&-2&11&8&-12&0\\& & 8& -76& \color{blue}{272} & \\ \hline &-2&19&\color{blue}{-68}&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -12 } + \color{orangered}{ 272 } = \color{orangered}{ 260 } $
$$ \begin{array}{c|rrrrr}-4&-2&11&8&\color{orangered}{ -12 }&0\\& & 8& -76& \color{orangered}{272} & \\ \hline &-2&19&-68&\color{orangered}{260}& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -4 } \cdot \color{blue}{ 260 } = \color{blue}{ -1040 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-4}&-2&11&8&-12&0\\& & 8& -76& 272& \color{blue}{-1040} \\ \hline &-2&19&-68&\color{blue}{260}& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ \left( -1040 \right) } = \color{orangered}{ -1040 } $
$$ \begin{array}{c|rrrrr}-4&-2&11&8&-12&\color{orangered}{ 0 }\\& & 8& -76& 272& \color{orangered}{-1040} \\ \hline &\color{blue}{-2}&\color{blue}{19}&\color{blue}{-68}&\color{blue}{260}&\color{orangered}{-1040} \end{array} $$Bottom line represents the quotient $ \color{blue}{ -2x^{3}+19x^{2}-68x+260 } $ with a remainder of $ \color{red}{ -1040 } $.