We can multiply polynomials by using a GRID METHOD

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

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

Fill in each empty box by multiplying the intersecting terms.

$$ \begin{darray}{|c|c|c|c|c|}\hline & \color{blue}{2x^3} & \color{blue}{5x^2} & \color{blue}{-3x} & \color{blue}{1} \\ \hline \color{blue}{2x} & \color{orangered}{4x^4} & \color{orangered}{10x^3} & \color{orangered}{-6x^2} & \color{orangered}{2x} \\ \hline \color{blue}{5} & \color{orangered}{10x^3} & \color{orangered}{25x^2} & \color{orangered}{-15x} & \color{orangered}{5} \\ \hline \end{darray} $$

Combine like terms:

$$ 4x^4 + 10x^3 + 10x^3-6x^2 + 25x^2 + 2x-15x + 5 = \\ 4x^4+20x^3+19x^2-13x+5 $$

Polynomial Operations Calculator