We can multiply polynomials by using a GRID METHOD

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

$$ \begin{darray}{|c|c|c|c|}\hline & \color{blue}{2m^2} & \color{blue}{-m} & \color{blue}{7} \\ \hline \color{blue}{3m^2} & & & \\ \hline \color{blue}{m} & & & \\ \hline \color{blue}{-5} & & & \\ \hline \end{darray} $$

Fill in each empty box by multiplying the intersecting terms.

$$ \begin{darray}{|c|c|c|c|}\hline & \color{blue}{2m^2} & \color{blue}{-m} & \color{blue}{7} \\ \hline \color{blue}{3m^2} & \color{orangered}{6m^4} & \color{orangered}{-3m^3} & \color{orangered}{21m^2} \\ \hline \color{blue}{m} & \color{orangered}{2m^3} & \color{orangered}{-m^2} & \color{orangered}{7m} \\ \hline \color{blue}{-5} & \color{orangered}{-10m^2} & \color{orangered}{5m} & \color{orangered}{-35} \\ \hline \end{darray} $$

Combine like terms:

$$ 6m^4-3m^3 + 2m^3 + 21m^2-m^2-10m^2 + 7m + 5m-35 = \\ 6m^4-m^3+10m^2+12m-35 $$

Polynomial Operations Calculator