We can multiply polynomials by using a GRID METHOD

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

$$ \begin{darray}{|c|c|c|c|}\hline & \color{blue}{x^3} & \color{blue}{x^2} & \color{blue}{1} \\ \hline \color{blue}{x^2} & & & \\ \hline \color{blue}{-x} & & & \\ \hline \color{blue}{-5} & & & \\ \hline \end{darray} $$

Fill in each empty box by multiplying the intersecting terms.

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

Combine like terms:

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

Polynomial Operations Calculator