We can multiply polynomials by using a GRID METHOD

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

$$ \begin{darray}{|c|c|c|c|}\hline & \color{blue}{5x^2} & \color{blue}{-6x} & \color{blue}{5} \\ \hline \color{blue}{2x^2} & & & \\ \hline \color{blue}{-12x} & & & \\ \hline \color{blue}{8} & & & \\ \hline \end{darray} $$

Fill in each empty box by multiplying the intersecting terms.

$$ \begin{darray}{|c|c|c|c|}\hline & \color{blue}{5x^2} & \color{blue}{-6x} & \color{blue}{5} \\ \hline \color{blue}{2x^2} & \color{orangered}{10x^4} & \color{orangered}{-12x^3} & \color{orangered}{10x^2} \\ \hline \color{blue}{-12x} & \color{orangered}{-60x^3} & \color{orangered}{72x^2} & \color{orangered}{-60x} \\ \hline \color{blue}{8} & \color{orangered}{40x^2} & \color{orangered}{-48x} & \color{orangered}{40} \\ \hline \end{darray} $$

Combine like terms:

$$ 10x^4-12x^3-60x^3 + 10x^2 + 72x^2 + 40x^2-60x-48x + 40 = \\ 10x^4-72x^3+122x^2-108x+40 $$

Polynomial Operations Calculator