We can multiply polynomials by using a GRID METHOD

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

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

Fill in each empty box by multiplying the intersecting terms.

$$ \begin{darray}{|c|c|c|c|}\hline & \color{blue}{4x^2} & \color{blue}{-5x} & \color{blue}{5} \\ \hline \color{blue}{-2x^2} & \color{orangered}{-8x^4} & \color{orangered}{10x^3} & \color{orangered}{-10x^2} \\ \hline \color{blue}{-x} & \color{orangered}{-4x^3} & \color{orangered}{5x^2} & \color{orangered}{-5x} \\ \hline \color{blue}{-2} & \color{orangered}{-8x^2} & \color{orangered}{10x} & \color{orangered}{-10} \\ \hline \end{darray} $$

Combine like terms:

$$ -8x^4 + 10x^3-4x^3-10x^2 + 5x^2-8x^2-5x + 10x-10 = \\ -8x^4+6x^3-13x^2+5x-10 $$

Polynomial Operations Calculator