Divide using synthetic division $$ ( 9x^{3}-39x^{2}-29x-5 ) \div ( 3x+1 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}-1&9&-39&-29&-5\\& & -9& 48& \color{black}{-19} \\ \hline &\color{blue}{9}&\color{blue}{-48}&\color{blue}{19}&\color{orangered}{-24} \end{array} $$The solution is:
$$ \frac{ 9x^{3}-39x^{2}-29x-5 }{ x+1 } = \color{blue}{9x^{2}-48x+19} \color{red}{~-~} \frac{ \color{red}{ 24 } }{ x+1 } $$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.
$$ \begin{array}{c|rrrr}\color{blue}{-1}&9&-39&-29&-5\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}-1&\color{orangered}{ 9 }&-39&-29&-5\\& & & & \\ \hline &\color{orangered}{9}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -1 } \cdot \color{blue}{ 9 } = \color{blue}{ -9 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-1}&9&-39&-29&-5\\& & \color{blue}{-9} & & \\ \hline &\color{blue}{9}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -39 } + \color{orangered}{ \left( -9 \right) } = \color{orangered}{ -48 } $
$$ \begin{array}{c|rrrr}-1&9&\color{orangered}{ -39 }&-29&-5\\& & \color{orangered}{-9} & & \\ \hline &9&\color{orangered}{-48}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -1 } \cdot \color{blue}{ \left( -48 \right) } = \color{blue}{ 48 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-1}&9&-39&-29&-5\\& & -9& \color{blue}{48} & \\ \hline &9&\color{blue}{-48}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -29 } + \color{orangered}{ 48 } = \color{orangered}{ 19 } $
$$ \begin{array}{c|rrrr}-1&9&-39&\color{orangered}{ -29 }&-5\\& & -9& \color{orangered}{48} & \\ \hline &9&-48&\color{orangered}{19}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -1 } \cdot \color{blue}{ 19 } = \color{blue}{ -19 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-1}&9&-39&-29&-5\\& & -9& 48& \color{blue}{-19} \\ \hline &9&-48&\color{blue}{19}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -5 } + \color{orangered}{ \left( -19 \right) } = \color{orangered}{ -24 } $
$$ \begin{array}{c|rrrr}-1&9&-39&-29&\color{orangered}{ -5 }\\& & -9& 48& \color{orangered}{-19} \\ \hline &\color{blue}{9}&\color{blue}{-48}&\color{blue}{19}&\color{orangered}{-24} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 9x^{2}-48x+19 } $ with a remainder of $ \color{red}{ -24 } $.