Divide using synthetic division $$ ( x^{3}+2x^{2}-7x-21 ) \div ( x+3 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}-3&1&2&-7&-21\\& & -3& 3& \color{black}{12} \\ \hline &\color{blue}{1}&\color{blue}{-1}&\color{blue}{-4}&\color{orangered}{-9} \end{array} $$The solution is:
$$ \frac{ x^{3}+2x^{2}-7x-21 }{ x+3 } = \color{blue}{x^{2}-x-4} \color{red}{~-~} \frac{ \color{red}{ 9 } }{ x+3 } $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x + 3 = 0 $ ( $ x = \color{blue}{ -3 } $ ) at the left.
$$ \begin{array}{c|rrrr}\color{blue}{-3}&1&2&-7&-21\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}-3&\color{orangered}{ 1 }&2&-7&-21\\& & & & \\ \hline &\color{orangered}{1}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -3 } \cdot \color{blue}{ 1 } = \color{blue}{ -3 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-3}&1&2&-7&-21\\& & \color{blue}{-3} & & \\ \hline &\color{blue}{1}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 2 } + \color{orangered}{ \left( -3 \right) } = \color{orangered}{ -1 } $
$$ \begin{array}{c|rrrr}-3&1&\color{orangered}{ 2 }&-7&-21\\& & \color{orangered}{-3} & & \\ \hline &1&\color{orangered}{-1}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -3 } \cdot \color{blue}{ \left( -1 \right) } = \color{blue}{ 3 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-3}&1&2&-7&-21\\& & -3& \color{blue}{3} & \\ \hline &1&\color{blue}{-1}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -7 } + \color{orangered}{ 3 } = \color{orangered}{ -4 } $
$$ \begin{array}{c|rrrr}-3&1&2&\color{orangered}{ -7 }&-21\\& & -3& \color{orangered}{3} & \\ \hline &1&-1&\color{orangered}{-4}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -3 } \cdot \color{blue}{ \left( -4 \right) } = \color{blue}{ 12 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-3}&1&2&-7&-21\\& & -3& 3& \color{blue}{12} \\ \hline &1&-1&\color{blue}{-4}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -21 } + \color{orangered}{ 12 } = \color{orangered}{ -9 } $
$$ \begin{array}{c|rrrr}-3&1&2&-7&\color{orangered}{ -21 }\\& & -3& 3& \color{orangered}{12} \\ \hline &\color{blue}{1}&\color{blue}{-1}&\color{blue}{-4}&\color{orangered}{-9} \end{array} $$Bottom line represents the quotient $ \color{blue}{ x^{2}-x-4 } $ with a remainder of $ \color{red}{ -9 } $.