We can multiply polynomials by using a GRID METHOD

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

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

Fill in each empty box by multiplying the intersecting terms.

$$ \begin{darray}{|c|c|c|c|}\hline & \color{blue}{x^2} & \color{blue}{2x} & \color{blue}{-2} \\ \hline \color{blue}{3x^2} & \color{orangered}{3x^4} & \color{orangered}{6x^3} & \color{orangered}{-6x^2} \\ \hline \color{blue}{-x} & \color{orangered}{-x^3} & \color{orangered}{-2x^2} & \color{orangered}{2x} \\ \hline \color{blue}{-9} & \color{orangered}{-9x^2} & \color{orangered}{-18x} & \color{orangered}{18} \\ \hline \end{darray} $$

Combine like terms:

$$ 3x^4 + 6x^3-x^3-6x^2-2x^2-9x^2 + 2x-18x + 18 = \\ 3x^4+5x^3-17x^2-16x+18 $$

Polynomial Operations Calculator