We can multiply polynomials by using a GRID METHOD

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

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

Fill in each empty box by multiplying the intersecting terms.

$$ \begin{darray}{|c|c|c|c|}\hline & \color{blue}{x^2} & \color{blue}{-x} & \color{blue}{3} \\ \hline \color{blue}{x^3} & \color{orangered}{x^5} & \color{orangered}{-x^4} & \color{orangered}{3x^3} \\ \hline \color{blue}{-x} & \color{orangered}{-x^3} & \color{orangered}{x^2} & \color{orangered}{-3x} \\ \hline \color{blue}{3} & \color{orangered}{3x^2} & \color{orangered}{-3x} & \color{orangered}{9} \\ \hline \end{darray} $$

Combine like terms:

$$ x^5-x^4-x^3 + 3x^3 + x^2 + 3x^2-3x-3x + 9 = \\ x^5-x^4+2x^3+4x^2-6x+9 $$

Polynomial Operations Calculator