We can multiply polynomials by using a GRID METHOD

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

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

Fill in each empty box by multiplying the intersecting terms.

$$ \begin{darray}{|c|c|c|c|}\hline & \color{blue}{x^2} & \color{blue}{x} & \color{blue}{-6} \\ \hline \color{blue}{-x^2} & \color{orangered}{-x^4} & \color{orangered}{-x^3} & \color{orangered}{6x^2} \\ \hline \color{blue}{x} & \color{orangered}{x^3} & \color{orangered}{x^2} & \color{orangered}{-6x} \\ \hline \color{blue}{3} & \color{orangered}{3x^2} & \color{orangered}{3x} & \color{orangered}{-18} \\ \hline \end{darray} $$

Combine like terms:

$$ -x^4-x^3 + x^3 + 6x^2 + x^2 + 3x^2-6x + 3x-18 = \\ -x^4+10x^2-3x-18 $$

Polynomial Operations Calculator