Divide using synthetic division $$ ( 5x^{3}+49x^{2}+35x-29 ) \div ( x+9 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}-9&5&49&35&-29\\& & -45& -36& \color{black}{9} \\ \hline &\color{blue}{5}&\color{blue}{4}&\color{blue}{-1}&\color{orangered}{-20} \end{array} $$The solution is:
$$ \frac{ 5x^{3}+49x^{2}+35x-29 }{ x+9 } = \color{blue}{5x^{2}+4x-1} \color{red}{~-~} \frac{ \color{red}{ 20 } }{ x+9 } $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x + 9 = 0 $ ( $ x = \color{blue}{ -9 } $ ) at the left.
$$ \begin{array}{c|rrrr}\color{blue}{-9}&5&49&35&-29\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}-9&\color{orangered}{ 5 }&49&35&-29\\& & & & \\ \hline &\color{orangered}{5}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -9 } \cdot \color{blue}{ 5 } = \color{blue}{ -45 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-9}&5&49&35&-29\\& & \color{blue}{-45} & & \\ \hline &\color{blue}{5}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 49 } + \color{orangered}{ \left( -45 \right) } = \color{orangered}{ 4 } $
$$ \begin{array}{c|rrrr}-9&5&\color{orangered}{ 49 }&35&-29\\& & \color{orangered}{-45} & & \\ \hline &5&\color{orangered}{4}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -9 } \cdot \color{blue}{ 4 } = \color{blue}{ -36 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-9}&5&49&35&-29\\& & -45& \color{blue}{-36} & \\ \hline &5&\color{blue}{4}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 35 } + \color{orangered}{ \left( -36 \right) } = \color{orangered}{ -1 } $
$$ \begin{array}{c|rrrr}-9&5&49&\color{orangered}{ 35 }&-29\\& & -45& \color{orangered}{-36} & \\ \hline &5&4&\color{orangered}{-1}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -9 } \cdot \color{blue}{ \left( -1 \right) } = \color{blue}{ 9 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-9}&5&49&35&-29\\& & -45& -36& \color{blue}{9} \\ \hline &5&4&\color{blue}{-1}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -29 } + \color{orangered}{ 9 } = \color{orangered}{ -20 } $
$$ \begin{array}{c|rrrr}-9&5&49&35&\color{orangered}{ -29 }\\& & -45& -36& \color{orangered}{9} \\ \hline &\color{blue}{5}&\color{blue}{4}&\color{blue}{-1}&\color{orangered}{-20} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 5x^{2}+4x-1 } $ with a remainder of $ \color{red}{ -20 } $.