We can multiply polynomials by using a GRID METHOD

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

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

Fill in each empty box by multiplying the intersecting terms.

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

Combine like terms:

$$ 10x^4 + 15x^3-2x^3-5x^2-3x^2 + 4x^2 + x + 6x-2 = \\ 10x^4+13x^3-4x^2+7x-2 $$

Polynomial Operations Calculator