We can multiply polynomials by using a GRID METHOD

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

$$ \begin{darray}{|c|c|c|c|}\hline & \color{blue}{4x^2} & \color{blue}{-x} & \color{blue}{16} \\ \hline \color{blue}{x^2} & & & \\ \hline \color{blue}{3x} & & & \\ \hline \color{blue}{4} & & & \\ \hline \end{darray} $$

Fill in each empty box by multiplying the intersecting terms.

$$ \begin{darray}{|c|c|c|c|}\hline & \color{blue}{4x^2} & \color{blue}{-x} & \color{blue}{16} \\ \hline \color{blue}{x^2} & \color{orangered}{4x^4} & \color{orangered}{-x^3} & \color{orangered}{16x^2} \\ \hline \color{blue}{3x} & \color{orangered}{12x^3} & \color{orangered}{-3x^2} & \color{orangered}{48x} \\ \hline \color{blue}{4} & \color{orangered}{16x^2} & \color{orangered}{-4x} & \color{orangered}{64} \\ \hline \end{darray} $$

Combine like terms:

$$ 4x^4-x^3 + 12x^3 + 16x^2-3x^2 + 16x^2 + 48x-4x + 64 = \\ 4x^4+11x^3+29x^2+44x+64 $$

Polynomial Operations Calculator