We can multiply polynomials by using a GRID METHOD

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

$$ \begin{darray}{|c|c|c|c|c|}\hline & \color{blue}{x^3} & \color{blue}{-12x^2} & \color{blue}{20x} & \color{blue}{90} \\ \hline \color{blue}{x} & & & & \\ \hline \color{blue}{-6} & & & & \\ \hline \end{darray} $$

Fill in each empty box by multiplying the intersecting terms.

$$ \begin{darray}{|c|c|c|c|c|}\hline & \color{blue}{x^3} & \color{blue}{-12x^2} & \color{blue}{20x} & \color{blue}{90} \\ \hline \color{blue}{x} & \color{orangered}{x^4} & \color{orangered}{-12x^3} & \color{orangered}{20x^2} & \color{orangered}{90x} \\ \hline \color{blue}{-6} & \color{orangered}{-6x^3} & \color{orangered}{72x^2} & \color{orangered}{-120x} & \color{orangered}{-540} \\ \hline \end{darray} $$

Combine like terms:

$$ x^4-12x^3-6x^3 + 20x^2 + 72x^2 + 90x-120x-540 = \\ x^4-18x^3+92x^2-30x-540 $$

Polynomial Operations Calculator