We can multiply polynomials by using a GRID METHOD

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

$$ \begin{darray}{|c|c|c|c|}\hline & \color{blue}{4x^2} & \color{blue}{19x} & \color{blue}{12} \\ \hline \color{blue}{x^2} & & & \\ \hline \color{blue}{-2x} & & & \\ \hline \color{blue}{1} & & & \\ \hline \end{darray} $$

Fill in each empty box by multiplying the intersecting terms.

$$ \begin{darray}{|c|c|c|c|}\hline & \color{blue}{4x^2} & \color{blue}{19x} & \color{blue}{12} \\ \hline \color{blue}{x^2} & \color{orangered}{4x^4} & \color{orangered}{19x^3} & \color{orangered}{12x^2} \\ \hline \color{blue}{-2x} & \color{orangered}{-8x^3} & \color{orangered}{-38x^2} & \color{orangered}{-24x} \\ \hline \color{blue}{1} & \color{orangered}{4x^2} & \color{orangered}{19x} & \color{orangered}{12} \\ \hline \end{darray} $$

Combine like terms:

$$ 4x^4 + 19x^3-8x^3 + 12x^2-38x^2 + 4x^2-24x + 19x + 12 = \\ 4x^4+11x^3-22x^2-5x+12 $$

Polynomial Operations Calculator