Divide using synthetic division $$ ( 3x^{3}-11x^{2}-30x+29 ) \div ( 3x+7 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}-7&3&-11&-30&29\\& & -21& 224& \color{black}{-1358} \\ \hline &\color{blue}{3}&\color{blue}{-32}&\color{blue}{194}&\color{orangered}{-1329} \end{array} $$The solution is:
$$ \frac{ 3x^{3}-11x^{2}-30x+29 }{ x+7 } = \color{blue}{3x^{2}-32x+194} \color{red}{~-~} \frac{ \color{red}{ 1329 } }{ x+7 } $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x + 7 = 0 $ ( $ x = \color{blue}{ -7 } $ ) at the left.
$$ \begin{array}{c|rrrr}\color{blue}{-7}&3&-11&-30&29\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}-7&\color{orangered}{ 3 }&-11&-30&29\\& & & & \\ \hline &\color{orangered}{3}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -7 } \cdot \color{blue}{ 3 } = \color{blue}{ -21 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-7}&3&-11&-30&29\\& & \color{blue}{-21} & & \\ \hline &\color{blue}{3}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -11 } + \color{orangered}{ \left( -21 \right) } = \color{orangered}{ -32 } $
$$ \begin{array}{c|rrrr}-7&3&\color{orangered}{ -11 }&-30&29\\& & \color{orangered}{-21} & & \\ \hline &3&\color{orangered}{-32}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -7 } \cdot \color{blue}{ \left( -32 \right) } = \color{blue}{ 224 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-7}&3&-11&-30&29\\& & -21& \color{blue}{224} & \\ \hline &3&\color{blue}{-32}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -30 } + \color{orangered}{ 224 } = \color{orangered}{ 194 } $
$$ \begin{array}{c|rrrr}-7&3&-11&\color{orangered}{ -30 }&29\\& & -21& \color{orangered}{224} & \\ \hline &3&-32&\color{orangered}{194}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -7 } \cdot \color{blue}{ 194 } = \color{blue}{ -1358 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-7}&3&-11&-30&29\\& & -21& 224& \color{blue}{-1358} \\ \hline &3&-32&\color{blue}{194}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 29 } + \color{orangered}{ \left( -1358 \right) } = \color{orangered}{ -1329 } $
$$ \begin{array}{c|rrrr}-7&3&-11&-30&\color{orangered}{ 29 }\\& & -21& 224& \color{orangered}{-1358} \\ \hline &\color{blue}{3}&\color{blue}{-32}&\color{blue}{194}&\color{orangered}{-1329} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 3x^{2}-32x+194 } $ with a remainder of $ \color{red}{ -1329 } $.