We can multiply polynomials by using a GRID METHOD

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

$$ \begin{darray}{|c|c|c|c|}\hline & \color{blue}{-x^2} & \color{blue}{4x} & \color{blue}{3} \\ \hline \color{blue}{3x^2} & & & \\ \hline \color{blue}{-7x} & & & \\ \hline \color{blue}{6} & & & \\ \hline \end{darray} $$

Fill in each empty box by multiplying the intersecting terms.

$$ \begin{darray}{|c|c|c|c|}\hline & \color{blue}{-x^2} & \color{blue}{4x} & \color{blue}{3} \\ \hline \color{blue}{3x^2} & \color{orangered}{-3x^4} & \color{orangered}{12x^3} & \color{orangered}{9x^2} \\ \hline \color{blue}{-7x} & \color{orangered}{7x^3} & \color{orangered}{-28x^2} & \color{orangered}{-21x} \\ \hline \color{blue}{6} & \color{orangered}{-6x^2} & \color{orangered}{24x} & \color{orangered}{18} \\ \hline \end{darray} $$

Combine like terms:

$$ -3x^4 + 12x^3 + 7x^3 + 9x^2-28x^2-6x^2-21x + 24x + 18 = \\ -3x^4+19x^3-25x^2+3x+18 $$

Polynomial Operations Calculator