We can multiply polynomials by using a GRID METHOD

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

$$ \begin{darray}{|c|c|c|c|}\hline & \color{blue}{-2x^2} & \color{blue}{2x} & \color{blue}{-4} \\ \hline \color{blue}{x^2} & & & \\ \hline \color{blue}{-x} & & & \\ \hline \color{blue}{-5} & & & \\ \hline \end{darray} $$

Fill in each empty box by multiplying the intersecting terms.

$$ \begin{darray}{|c|c|c|c|}\hline & \color{blue}{-2x^2} & \color{blue}{2x} & \color{blue}{-4} \\ \hline \color{blue}{x^2} & \color{orangered}{-2x^4} & \color{orangered}{2x^3} & \color{orangered}{-4x^2} \\ \hline \color{blue}{-x} & \color{orangered}{2x^3} & \color{orangered}{-2x^2} & \color{orangered}{4x} \\ \hline \color{blue}{-5} & \color{orangered}{10x^2} & \color{orangered}{-10x} & \color{orangered}{20} \\ \hline \end{darray} $$

Combine like terms:

$$ -2x^4 + 2x^3 + 2x^3-4x^2-2x^2 + 10x^2 + 4x-10x + 20 = \\ -2x^4+4x^3+4x^2-6x+20 $$

Polynomial Operations Calculator