We can multiply polynomials by using a GRID METHOD

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

$$ \begin{darray}{|c|c|c|c|}\hline & \color{blue}{-3v^2} & \color{blue}{6v} & \color{blue}{-6} \\ \hline \color{blue}{-5v^2} & & & \\ \hline \color{blue}{5v} & & & \\ \hline \color{blue}{7} & & & \\ \hline \end{darray} $$

Fill in each empty box by multiplying the intersecting terms.

$$ \begin{darray}{|c|c|c|c|}\hline & \color{blue}{-3v^2} & \color{blue}{6v} & \color{blue}{-6} \\ \hline \color{blue}{-5v^2} & \color{orangered}{15v^4} & \color{orangered}{-30v^3} & \color{orangered}{30v^2} \\ \hline \color{blue}{5v} & \color{orangered}{-15v^3} & \color{orangered}{30v^2} & \color{orangered}{-30v} \\ \hline \color{blue}{7} & \color{orangered}{-21v^2} & \color{orangered}{42v} & \color{orangered}{-42} \\ \hline \end{darray} $$

Combine like terms:

$$ 15v^4-30v^3-15v^3 + 30v^2 + 30v^2-21v^2-30v + 42v-42 = \\ 15v^4-45v^3+39v^2+12v-42 $$

Polynomial Operations Calculator