We can multiply polynomials by using a GRID METHOD

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

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

Fill in each empty box by multiplying the intersecting terms.

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

Combine like terms:

$$ -6x^4 + 10x^3-9x^3-6x^2 + 15x^2 + 6x^2-9x-10x + 6 = \\ -6x^4+x^3+15x^2-19x+6 $$

Polynomial Operations Calculator