We can multiply polynomials by using a GRID METHOD

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

$$ \begin{darray}{|c|c|c|c|}\hline & \color{blue}{6v^2} & \color{blue}{-v} & \color{blue}{-4} \\ \hline \color{blue}{2v^2} & & & \\ \hline \color{blue}{-7v} & & & \\ \hline \color{blue}{-6} & & & \\ \hline \end{darray} $$

Fill in each empty box by multiplying the intersecting terms.

$$ \begin{darray}{|c|c|c|c|}\hline & \color{blue}{6v^2} & \color{blue}{-v} & \color{blue}{-4} \\ \hline \color{blue}{2v^2} & \color{orangered}{12v^4} & \color{orangered}{-2v^3} & \color{orangered}{-8v^2} \\ \hline \color{blue}{-7v} & \color{orangered}{-42v^3} & \color{orangered}{7v^2} & \color{orangered}{28v} \\ \hline \color{blue}{-6} & \color{orangered}{-36v^2} & \color{orangered}{6v} & \color{orangered}{24} \\ \hline \end{darray} $$

Combine like terms:

$$ 12v^4-2v^3-42v^3-8v^2 + 7v^2-36v^2 + 28v + 6v + 24 = \\ 12v^4-44v^3-37v^2+34v+24 $$

Polynomial Operations Calculator