We can multiply polynomials by using a GRID METHOD

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

$$ \begin{darray}{|c|c|c|c|c|}\hline & \color{blue}{15x^3} & \color{blue}{4x^2} & \color{blue}{-x} & \color{blue}{3} \\ \hline \color{blue}{7x^3} & & & & \\ \hline \color{blue}{-1} & & & & \\ \hline \end{darray} $$

Fill in each empty box by multiplying the intersecting terms.

$$ \begin{darray}{|c|c|c|c|c|}\hline & \color{blue}{15x^3} & \color{blue}{4x^2} & \color{blue}{-x} & \color{blue}{3} \\ \hline \color{blue}{7x^3} & \color{orangered}{105x^6} & \color{orangered}{28x^5} & \color{orangered}{-7x^4} & \color{orangered}{21x^3} \\ \hline \color{blue}{-1} & \color{orangered}{-15x^3} & \color{orangered}{-4x^2} & \color{orangered}{x} & \color{orangered}{-3} \\ \hline \end{darray} $$

Combine like terms:

$$ 105x^6 + 28x^5-15x^3-7x^4-4x^2 + 21x^3 + x-3 = \\ 105x^6+28x^5-7x^4+6x^3-4x^2+x-3 $$

Polynomial Operations Calculator