We can multiply polynomials by using a GRID METHOD

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

$$ \begin{darray}{|c|c|c|c|}\hline & \color{blue}{9x^2} & \color{blue}{-9x} & \color{blue}{-7} \\ \hline \color{blue}{-x^2} & & & \\ \hline \color{blue}{6x} & & & \\ \hline \color{blue}{2} & & & \\ \hline \end{darray} $$

Fill in each empty box by multiplying the intersecting terms.

$$ \begin{darray}{|c|c|c|c|}\hline & \color{blue}{9x^2} & \color{blue}{-9x} & \color{blue}{-7} \\ \hline \color{blue}{-x^2} & \color{orangered}{-9x^4} & \color{orangered}{9x^3} & \color{orangered}{7x^2} \\ \hline \color{blue}{6x} & \color{orangered}{54x^3} & \color{orangered}{-54x^2} & \color{orangered}{-42x} \\ \hline \color{blue}{2} & \color{orangered}{18x^2} & \color{orangered}{-18x} & \color{orangered}{-14} \\ \hline \end{darray} $$

Combine like terms:

$$ -9x^4 + 9x^3 + 54x^3 + 7x^2-54x^2 + 18x^2-42x-18x-14 = \\ -9x^4+63x^3-29x^2-60x-14 $$

Polynomial Operations Calculator