We can multiply polynomials by using a GRID METHOD

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

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

Fill in each empty box by multiplying the intersecting terms.

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

Combine like terms:

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

Polynomial Operations Calculator