We can multiply polynomials by using a GRID METHOD

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

$$ \begin{darray}{|c|c|c|c|c|}\hline & \color{blue}{3x^3} & \color{blue}{-2x^2} & \color{blue}{9x} & \color{blue}{-6} \\ \hline \color{blue}{2x} & & & & \\ \hline \color{blue}{5} & & & & \\ \hline \end{darray} $$

Fill in each empty box by multiplying the intersecting terms.

$$ \begin{darray}{|c|c|c|c|c|}\hline & \color{blue}{3x^3} & \color{blue}{-2x^2} & \color{blue}{9x} & \color{blue}{-6} \\ \hline \color{blue}{2x} & \color{orangered}{6x^4} & \color{orangered}{-4x^3} & \color{orangered}{18x^2} & \color{orangered}{-12x} \\ \hline \color{blue}{5} & \color{orangered}{15x^3} & \color{orangered}{-10x^2} & \color{orangered}{45x} & \color{orangered}{-30} \\ \hline \end{darray} $$

Combine like terms:

$$ 6x^4-4x^3 + 15x^3 + 18x^2-10x^2-12x + 45x-30 = \\ 6x^4+11x^3+8x^2+33x-30 $$

Polynomial Operations Calculator