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