We can multiply polynomials by using a GRID METHOD

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

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

Fill in each empty box by multiplying the intersecting terms.

$$ \begin{darray}{|c|c|c|c|}\hline & \color{blue}{-7x^2} & \color{blue}{-3x} & \color{blue}{5} \\ \hline \color{blue}{7x^2} & \color{orangered}{-49x^4} & \color{orangered}{-21x^3} & \color{orangered}{35x^2} \\ \hline \color{blue}{-6x} & \color{orangered}{42x^3} & \color{orangered}{18x^2} & \color{orangered}{-30x} \\ \hline \color{blue}{7} & \color{orangered}{-49x^2} & \color{orangered}{-21x} & \color{orangered}{35} \\ \hline \end{darray} $$

Combine like terms:

$$ -49x^4-21x^3 + 42x^3 + 35x^2 + 18x^2-49x^2-30x-21x + 35 = \\ -49x^4+21x^3+4x^2-51x+35 $$

Polynomial Operations Calculator