We can multiply polynomials by using a GRID METHOD

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

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

Fill in each empty box by multiplying the intersecting terms.

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

Combine like terms:

$$ x^5 + 5x^4-x^4-3x^3-5x^3-5x^2 + 3x^2 + 2x + 5x-2 = \\ x^5+4x^4-8x^3-2x^2+7x-2 $$

Polynomial Operations Calculator