We can multiply polynomials by using a GRID METHOD

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

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

Fill in each empty box by multiplying the intersecting terms.

$$ \begin{darray}{|c|c|c|c|}\hline & \color{blue}{5x^2} & \color{blue}{2x} & \color{blue}{-4} \\ \hline \color{blue}{8x^2} & \color{orangered}{40x^4} & \color{orangered}{16x^3} & \color{orangered}{-32x^2} \\ \hline \color{blue}{4x} & \color{orangered}{20x^3} & \color{orangered}{8x^2} & \color{orangered}{-16x} \\ \hline \color{blue}{-5} & \color{orangered}{-25x^2} & \color{orangered}{-10x} & \color{orangered}{20} \\ \hline \end{darray} $$

Combine like terms:

$$ 40x^4 + 16x^3 + 20x^3-32x^2 + 8x^2-25x^2-16x-10x + 20 = \\ 40x^4+36x^3-49x^2-26x+20 $$

Polynomial Operations Calculator