Divide using synthetic division $$ ( 15x^{2}+9 ) \div ( x+6 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrr}-6&15&0&9\\& & -90& \color{black}{540} \\ \hline &\color{blue}{15}&\color{blue}{-90}&\color{orangered}{549} \end{array} $$The solution is:
$$ \frac{ 15x^{2}+9 }{ x+6 } = \color{blue}{15x-90} ~+~ \frac{ \color{red}{ 549 } }{ x+6 } $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x + 6 = 0 $ ( $ x = \color{blue}{ -6 } $ ) at the left.
$$ \begin{array}{c|rrr}\color{blue}{-6}&15&0&9\\& & & \\ \hline &&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrr}-6&\color{orangered}{ 15 }&0&9\\& & & \\ \hline &\color{orangered}{15}&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -6 } \cdot \color{blue}{ 15 } = \color{blue}{ -90 } $.
$$ \begin{array}{c|rrr}\color{blue}{-6}&15&0&9\\& & \color{blue}{-90} & \\ \hline &\color{blue}{15}&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ \left( -90 \right) } = \color{orangered}{ -90 } $
$$ \begin{array}{c|rrr}-6&15&\color{orangered}{ 0 }&9\\& & \color{orangered}{-90} & \\ \hline &15&\color{orangered}{-90}& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -6 } \cdot \color{blue}{ \left( -90 \right) } = \color{blue}{ 540 } $.
$$ \begin{array}{c|rrr}\color{blue}{-6}&15&0&9\\& & -90& \color{blue}{540} \\ \hline &15&\color{blue}{-90}& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 9 } + \color{orangered}{ 540 } = \color{orangered}{ 549 } $
$$ \begin{array}{c|rrr}-6&15&0&\color{orangered}{ 9 }\\& & -90& \color{orangered}{540} \\ \hline &\color{blue}{15}&\color{blue}{-90}&\color{orangered}{549} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 15x-90 } $ with a remainder of $ \color{red}{ 549 } $.