We can multiply polynomials by using a GRID METHOD

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

$$ \begin{darray}{|c|c|c|c|}\hline & \color{blue}{3x^2} & \color{blue}{x} & \color{blue}{-2} \\ \hline \color{blue}{-4x^2} & & & \\ \hline \color{blue}{-2x} & & & \\ \hline \color{blue}{-1} & & & \\ \hline \end{darray} $$

Fill in each empty box by multiplying the intersecting terms.

$$ \begin{darray}{|c|c|c|c|}\hline & \color{blue}{3x^2} & \color{blue}{x} & \color{blue}{-2} \\ \hline \color{blue}{-4x^2} & \color{orangered}{-12x^4} & \color{orangered}{-4x^3} & \color{orangered}{8x^2} \\ \hline \color{blue}{-2x} & \color{orangered}{-6x^3} & \color{orangered}{-2x^2} & \color{orangered}{4x} \\ \hline \color{blue}{-1} & \color{orangered}{-3x^2} & \color{orangered}{-x} & \color{orangered}{2} \\ \hline \end{darray} $$

Combine like terms:

$$ -12x^4-4x^3-6x^3 + 8x^2-2x^2-3x^2 + 4x-x + 2 = \\ -12x^4-10x^3+3x^2+3x+2 $$

Polynomial Operations Calculator