Divide using synthetic division $$ ( -2x^{4}-x^{3}-3 ) \div ( x-3 ) $$
The synthetic division table is:
$$ \begin{array}{c|rrrrr}3&-2&-1&0&0&-3\\& & -6& -21& -63& \color{black}{-189} \\ \hline &\color{blue}{-2}&\color{blue}{-7}&\color{blue}{-21}&\color{blue}{-63}&\color{orangered}{-192} \end{array} $$The solution is:
$$ \dfrac{ -2x^{4}-x^{3}-3 }{ x-3 } = \color{blue}{-2x^{3}-7x^{2}-21x-63} \color{red}{~-~} \dfrac{ \color{red}{ 192 } }{ x-3 } $$Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x -3 = 0 $ ( $ x = \color{blue}{ 3 } $ ) at the left.
$$ \begin{array}{c|rrrrr}\color{blue}{3}&-2&-1&0&0&-3\\& & & & & \\ \hline &&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrr}3&\color{orangered}{ -2 }&-1&0&0&-3\\& & & & & \\ \hline &\color{orangered}{-2}&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 3 } \cdot \color{blue}{ \left( -2 \right) } = \color{blue}{ -6 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{3}&-2&-1&0&0&-3\\& & \color{blue}{-6} & & & \\ \hline &\color{blue}{-2}&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -1 } + \color{orangered}{ \left( -6 \right) } = \color{orangered}{ -7 } $
$$ \begin{array}{c|rrrrr}3&-2&\color{orangered}{ -1 }&0&0&-3\\& & \color{orangered}{-6} & & & \\ \hline &-2&\color{orangered}{-7}&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 3 } \cdot \color{blue}{ \left( -7 \right) } = \color{blue}{ -21 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{3}&-2&-1&0&0&-3\\& & -6& \color{blue}{-21} & & \\ \hline &-2&\color{blue}{-7}&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ \left( -21 \right) } = \color{orangered}{ -21 } $
$$ \begin{array}{c|rrrrr}3&-2&-1&\color{orangered}{ 0 }&0&-3\\& & -6& \color{orangered}{-21} & & \\ \hline &-2&-7&\color{orangered}{-21}&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 3 } \cdot \color{blue}{ \left( -21 \right) } = \color{blue}{ -63 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{3}&-2&-1&0&0&-3\\& & -6& -21& \color{blue}{-63} & \\ \hline &-2&-7&\color{blue}{-21}&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ \left( -63 \right) } = \color{orangered}{ -63 } $
$$ \begin{array}{c|rrrrr}3&-2&-1&0&\color{orangered}{ 0 }&-3\\& & -6& -21& \color{orangered}{-63} & \\ \hline &-2&-7&-21&\color{orangered}{-63}& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 3 } \cdot \color{blue}{ \left( -63 \right) } = \color{blue}{ -189 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{3}&-2&-1&0&0&-3\\& & -6& -21& -63& \color{blue}{-189} \\ \hline &-2&-7&-21&\color{blue}{-63}& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ -3 } + \color{orangered}{ \left( -189 \right) } = \color{orangered}{ -192 } $
$$ \begin{array}{c|rrrrr}3&-2&-1&0&0&\color{orangered}{ -3 }\\& & -6& -21& -63& \color{orangered}{-189} \\ \hline &\color{blue}{-2}&\color{blue}{-7}&\color{blue}{-21}&\color{blue}{-63}&\color{orangered}{-192} \end{array} $$Bottom line represents the quotient $ \color{blue}{ -2x^{3}-7x^{2}-21x-63 } $ with a remainder of $ \color{red}{ -192 } $.