We can multiply polynomials by using a GRID METHOD

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

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

Fill in each empty box by multiplying the intersecting terms.

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

Combine like terms:

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

Polynomial Operations Calculator