Multiply polynomial $ \, \color{blue}{ -x-6 } \, $ by $ \, \color{orangered}{ 3x^2+2x+9 }$.

$$ \left( \color{blue}{-x-6} \right) \cdot \left( \color{orangered}{3x^2+2x+9} \right) = -3x^3-20x^2-21x-54 $$

We can multiply polynomials by using a GRID METHOD

Write one of the polynomials across the top and the other down the left side.

$$ \begin{darray}{|c|c|c|c|}\hline & \color{blue}{3x^2} & \color{blue}{2x} & \color{blue}{9} \\ \hline \color{blue}{-x} & & & \\ \hline \color{blue}{-6} & & & \\ \hline \end{darray} $$

Fill in each empty box by multiplying the intersecting terms.

$$ \begin{darray}{|c|c|c|c|}\hline & \color{blue}{3x^2} & \color{blue}{2x} & \color{blue}{9} \\ \hline \color{blue}{-x} & \color{orangered}{-3x^3} & \color{orangered}{-2x^2} & \color{orangered}{-9x} \\ \hline \color{blue}{-6} & \color{orangered}{-18x^2} & \color{orangered}{-12x} & \color{orangered}{-54} \\ \hline \end{darray} $$

Combine like terms:

$$ -3x^3-2x^2-18x^2-9x-12x-54 = \\ -3x^3-20x^2-21x-54 $$

Polynomial Operations Calculator