We can multiply polynomials by using a GRID METHOD

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

$$ \begin{darray}{|c|c|c|c|c|}\hline & \color{blue}{x^3} & \color{blue}{3x^2} & \color{blue}{-4x} & \color{blue}{5} \\ \hline \color{blue}{x} & & & & \\ \hline \color{blue}{3} & & & & \\ \hline \end{darray} $$

Fill in each empty box by multiplying the intersecting terms.

$$ \begin{darray}{|c|c|c|c|c|}\hline & \color{blue}{x^3} & \color{blue}{3x^2} & \color{blue}{-4x} & \color{blue}{5} \\ \hline \color{blue}{x} & \color{orangered}{x^4} & \color{orangered}{3x^3} & \color{orangered}{-4x^2} & \color{orangered}{5x} \\ \hline \color{blue}{3} & \color{orangered}{3x^3} & \color{orangered}{9x^2} & \color{orangered}{-12x} & \color{orangered}{15} \\ \hline \end{darray} $$

Combine like terms:

$$ x^4 + 3x^3 + 3x^3-4x^2 + 9x^2 + 5x-12x + 15 = \\ x^4+6x^3+5x^2-7x+15 $$

Polynomial Operations Calculator