We can multiply polynomials by using a GRID METHOD

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

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

Fill in each empty box by multiplying the intersecting terms.

$$ \begin{darray}{|c|c|c|c|}\hline & \color{blue}{x^2} & \color{blue}{x} & \color{blue}{4} \\ \hline \color{blue}{2x^2} & \color{orangered}{2x^4} & \color{orangered}{2x^3} & \color{orangered}{8x^2} \\ \hline \color{blue}{-x} & \color{orangered}{-x^3} & \color{orangered}{-x^2} & \color{orangered}{-4x} \\ \hline \color{blue}{1} & \color{orangered}{x^2} & \color{orangered}{x} & \color{orangered}{4} \\ \hline \end{darray} $$

Combine like terms:

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

Polynomial Operations Calculator