Divide using synthetic division $$ ( 2x^{3}+6x^{2}-14x+9 ) \div ( x+3 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}-3&2&6&-14&9\\& & -6& 0& \color{black}{42} \\ \hline &\color{blue}{2}&\color{blue}{0}&\color{blue}{-14}&\color{orangered}{51} \end{array} $$The solution is:
$$ \frac{ 2x^{3}+6x^{2}-14x+9 }{ x+3 } = \color{blue}{2x^{2}-14} ~+~ \frac{ \color{red}{ 51 } }{ x+3 } $$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.
$$ \begin{array}{c|rrrr}\color{blue}{-3}&2&6&-14&9\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}-3&\color{orangered}{ 2 }&6&-14&9\\& & & & \\ \hline &\color{orangered}{2}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -3 } \cdot \color{blue}{ 2 } = \color{blue}{ -6 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-3}&2&6&-14&9\\& & \color{blue}{-6} & & \\ \hline &\color{blue}{2}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 6 } + \color{orangered}{ \left( -6 \right) } = \color{orangered}{ 0 } $
$$ \begin{array}{c|rrrr}-3&2&\color{orangered}{ 6 }&-14&9\\& & \color{orangered}{-6} & & \\ \hline &2&\color{orangered}{0}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -3 } \cdot \color{blue}{ 0 } = \color{blue}{ 0 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-3}&2&6&-14&9\\& & -6& \color{blue}{0} & \\ \hline &2&\color{blue}{0}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -14 } + \color{orangered}{ 0 } = \color{orangered}{ -14 } $
$$ \begin{array}{c|rrrr}-3&2&6&\color{orangered}{ -14 }&9\\& & -6& \color{orangered}{0} & \\ \hline &2&0&\color{orangered}{-14}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -3 } \cdot \color{blue}{ \left( -14 \right) } = \color{blue}{ 42 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-3}&2&6&-14&9\\& & -6& 0& \color{blue}{42} \\ \hline &2&0&\color{blue}{-14}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 9 } + \color{orangered}{ 42 } = \color{orangered}{ 51 } $
$$ \begin{array}{c|rrrr}-3&2&6&-14&\color{orangered}{ 9 }\\& & -6& 0& \color{orangered}{42} \\ \hline &\color{blue}{2}&\color{blue}{0}&\color{blue}{-14}&\color{orangered}{51} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 2x^{2}-14 } $ with a remainder of $ \color{red}{ 51 } $.