Divide using synthetic division $$ ( -3x^{2}-11x-3 ) \div ( x+1 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrr}-1&-3&-11&-3\\& & 3& \color{black}{8} \\ \hline &\color{blue}{-3}&\color{blue}{-8}&\color{orangered}{5} \end{array} $$The solution is:
$$ \frac{ -3x^{2}-11x-3 }{ x+1 } = \color{blue}{-3x-8} ~+~ \frac{ \color{red}{ 5 } }{ x+1 } $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x + 1 = 0 $ ( $ x = \color{blue}{ -1 } $ ) at the left.
$$ \begin{array}{c|rrr}\color{blue}{-1}&-3&-11&-3\\& & & \\ \hline &&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrr}-1&\color{orangered}{ -3 }&-11&-3\\& & & \\ \hline &\color{orangered}{-3}&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -1 } \cdot \color{blue}{ \left( -3 \right) } = \color{blue}{ 3 } $.
$$ \begin{array}{c|rrr}\color{blue}{-1}&-3&-11&-3\\& & \color{blue}{3} & \\ \hline &\color{blue}{-3}&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -11 } + \color{orangered}{ 3 } = \color{orangered}{ -8 } $
$$ \begin{array}{c|rrr}-1&-3&\color{orangered}{ -11 }&-3\\& & \color{orangered}{3} & \\ \hline &-3&\color{orangered}{-8}& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -1 } \cdot \color{blue}{ \left( -8 \right) } = \color{blue}{ 8 } $.
$$ \begin{array}{c|rrr}\color{blue}{-1}&-3&-11&-3\\& & 3& \color{blue}{8} \\ \hline &-3&\color{blue}{-8}& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -3 } + \color{orangered}{ 8 } = \color{orangered}{ 5 } $
$$ \begin{array}{c|rrr}-1&-3&-11&\color{orangered}{ -3 }\\& & 3& \color{orangered}{8} \\ \hline &\color{blue}{-3}&\color{blue}{-8}&\color{orangered}{5} \end{array} $$Bottom line represents the quotient $ \color{blue}{ -3x-8 } $ with a remainder of $ \color{red}{ 5 } $.