We can multiply polynomials by using a GRID METHOD

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

$$ \begin{darray}{|c|c|c|c|c|c|}\hline & \color{blue}{8x^4} & \color{blue}{-20x^3} & \color{blue}{-138x^2} & \color{blue}{401x} & \color{blue}{-164} \\ \hline \color{blue}{x} & & & & & \\ \hline \color{blue}{1} & & & & & \\ \hline \end{darray} $$

Fill in each empty box by multiplying the intersecting terms.

$$ \begin{darray}{|c|c|c|c|c|c|}\hline & \color{blue}{8x^4} & \color{blue}{-20x^3} & \color{blue}{-138x^2} & \color{blue}{401x} & \color{blue}{-164} \\ \hline \color{blue}{x} & \color{orangered}{8x^5} & \color{orangered}{-20x^4} & \color{orangered}{-138x^3} & \color{orangered}{401x^2} & \color{orangered}{-164x} \\ \hline \color{blue}{1} & \color{orangered}{8x^4} & \color{orangered}{-20x^3} & \color{orangered}{-138x^2} & \color{orangered}{401x} & \color{orangered}{-164} \\ \hline \end{darray} $$

Combine like terms:

$$ 8x^5-20x^4 + 8x^4-138x^3-20x^3 + 401x^2-138x^2-164x + 401x-164 = \\ 8x^5-12x^4-158x^3+263x^2+237x-164 $$

Polynomial Operations Calculator