We can multiply polynomials by using a GRID METHOD

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

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

Fill in each empty box by multiplying the intersecting terms.

$$ \begin{darray}{|c|c|c|c|}\hline & \color{blue}{2x^2} & \color{blue}{3x} & \color{blue}{-1} \\ \hline \color{blue}{-5x^2} & \color{orangered}{-10x^4} & \color{orangered}{-15x^3} & \color{orangered}{5x^2} \\ \hline \color{blue}{-2x} & \color{orangered}{-4x^3} & \color{orangered}{-6x^2} & \color{orangered}{2x} \\ \hline \color{blue}{4} & \color{orangered}{8x^2} & \color{orangered}{12x} & \color{orangered}{-4} \\ \hline \end{darray} $$

Combine like terms:

$$ -10x^4-15x^3-4x^3 + 5x^2-6x^2 + 8x^2 + 2x + 12x-4 = \\ -10x^4-19x^3+7x^2+14x-4 $$

Polynomial Operations Calculator