Divide using synthetic division $$ ( -3x^{3}-x^{2}-2 ) \div ( x-3 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}3&-3&-1&0&-2\\& & -9& -30& \color{black}{-90} \\ \hline &\color{blue}{-3}&\color{blue}{-10}&\color{blue}{-30}&\color{orangered}{-92} \end{array} $$The solution is:
$$ \frac{ -3x^{3}-x^{2}-2 }{ x-3 } = \color{blue}{-3x^{2}-10x-30} \color{red}{~-~} \frac{ \color{red}{ 92 } }{ x-3 } $$Explanation
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.
$$ \begin{array}{c|rrrr}\color{blue}{3}&-3&-1&0&-2\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}3&\color{orangered}{ -3 }&-1&0&-2\\& & & & \\ \hline &\color{orangered}{-3}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 3 } \cdot \color{blue}{ \left( -3 \right) } = \color{blue}{ -9 } $.
$$ \begin{array}{c|rrrr}\color{blue}{3}&-3&-1&0&-2\\& & \color{blue}{-9} & & \\ \hline &\color{blue}{-3}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -1 } + \color{orangered}{ \left( -9 \right) } = \color{orangered}{ -10 } $
$$ \begin{array}{c|rrrr}3&-3&\color{orangered}{ -1 }&0&-2\\& & \color{orangered}{-9} & & \\ \hline &-3&\color{orangered}{-10}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 3 } \cdot \color{blue}{ \left( -10 \right) } = \color{blue}{ -30 } $.
$$ \begin{array}{c|rrrr}\color{blue}{3}&-3&-1&0&-2\\& & -9& \color{blue}{-30} & \\ \hline &-3&\color{blue}{-10}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ \left( -30 \right) } = \color{orangered}{ -30 } $
$$ \begin{array}{c|rrrr}3&-3&-1&\color{orangered}{ 0 }&-2\\& & -9& \color{orangered}{-30} & \\ \hline &-3&-10&\color{orangered}{-30}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 3 } \cdot \color{blue}{ \left( -30 \right) } = \color{blue}{ -90 } $.
$$ \begin{array}{c|rrrr}\color{blue}{3}&-3&-1&0&-2\\& & -9& -30& \color{blue}{-90} \\ \hline &-3&-10&\color{blue}{-30}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -2 } + \color{orangered}{ \left( -90 \right) } = \color{orangered}{ -92 } $
$$ \begin{array}{c|rrrr}3&-3&-1&0&\color{orangered}{ -2 }\\& & -9& -30& \color{orangered}{-90} \\ \hline &\color{blue}{-3}&\color{blue}{-10}&\color{blue}{-30}&\color{orangered}{-92} \end{array} $$Bottom line represents the quotient $ \color{blue}{ -3x^{2}-10x-30 } $ with a remainder of $ \color{red}{ -92 } $.