We can multiply polynomials by using a GRID METHOD

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

$$ \begin{darray}{|c|c|c|c|}\hline & \color{blue}{x^2} & \color{blue}{12x} & \color{blue}{32} \\ \hline \color{blue}{x^2} & & & \\ \hline \color{blue}{8x} & & & \\ \hline \color{blue}{15} & & & \\ \hline \end{darray} $$

Fill in each empty box by multiplying the intersecting terms.

$$ \begin{darray}{|c|c|c|c|}\hline & \color{blue}{x^2} & \color{blue}{12x} & \color{blue}{32} \\ \hline \color{blue}{x^2} & \color{orangered}{x^4} & \color{orangered}{12x^3} & \color{orangered}{32x^2} \\ \hline \color{blue}{8x} & \color{orangered}{8x^3} & \color{orangered}{96x^2} & \color{orangered}{256x} \\ \hline \color{blue}{15} & \color{orangered}{15x^2} & \color{orangered}{180x} & \color{orangered}{480} \\ \hline \end{darray} $$

Combine like terms:

$$ x^4 + 12x^3 + 8x^3 + 32x^2 + 96x^2 + 15x^2 + 256x + 180x + 480 = \\ x^4+20x^3+143x^2+436x+480 $$

Polynomial Operations Calculator