Divide using synthetic division $$ ( x^{3}+9x^{2}+25x+21 ) \div ( x+3 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}-3&1&9&25&21\\& & -3& -18& \color{black}{-21} \\ \hline &\color{blue}{1}&\color{blue}{6}&\color{blue}{7}&\color{orangered}{0} \end{array} $$The solution is:
$$ \frac{ x^{3}+9x^{2}+25x+21 }{ x+3 } = \color{blue}{x^{2}+6x+7} $$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.
$$ \begin{array}{c|rrrr}\color{blue}{-3}&1&9&25&21\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}-3&\color{orangered}{ 1 }&9&25&21\\& & & & \\ \hline &\color{orangered}{1}&&& \end{array} $$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 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-3}&1&9&25&21\\& & \color{blue}{-3} & & \\ \hline &\color{blue}{1}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 9 } + \color{orangered}{ \left( -3 \right) } = \color{orangered}{ 6 } $
$$ \begin{array}{c|rrrr}-3&1&\color{orangered}{ 9 }&25&21\\& & \color{orangered}{-3} & & \\ \hline &1&\color{orangered}{6}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -3 } \cdot \color{blue}{ 6 } = \color{blue}{ -18 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-3}&1&9&25&21\\& & -3& \color{blue}{-18} & \\ \hline &1&\color{blue}{6}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 25 } + \color{orangered}{ \left( -18 \right) } = \color{orangered}{ 7 } $
$$ \begin{array}{c|rrrr}-3&1&9&\color{orangered}{ 25 }&21\\& & -3& \color{orangered}{-18} & \\ \hline &1&6&\color{orangered}{7}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -3 } \cdot \color{blue}{ 7 } = \color{blue}{ -21 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-3}&1&9&25&21\\& & -3& -18& \color{blue}{-21} \\ \hline &1&6&\color{blue}{7}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 21 } + \color{orangered}{ \left( -21 \right) } = \color{orangered}{ 0 } $
$$ \begin{array}{c|rrrr}-3&1&9&25&\color{orangered}{ 21 }\\& & -3& -18& \color{orangered}{-21} \\ \hline &\color{blue}{1}&\color{blue}{6}&\color{blue}{7}&\color{orangered}{0} \end{array} $$Bottom line represents the quotient $ \color{blue}{ x^{2}+6x+7 } $ with a remainder of $ \color{red}{ 0 } $.