We can multiply polynomials by using a GRID METHOD

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

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

Fill in each empty box by multiplying the intersecting terms.

$$ \begin{darray}{|c|c|c|c|c|}\hline & \color{blue}{x^3} & \color{blue}{x^2} & \color{blue}{-x} & \color{blue}{-5} \\ \hline \color{blue}{x^4} & \color{orangered}{x^7} & \color{orangered}{x^6} & \color{orangered}{-x^5} & \color{orangered}{-5x^4} \\ \hline \color{blue}{2} & \color{orangered}{2x^3} & \color{orangered}{2x^2} & \color{orangered}{-2x} & \color{orangered}{-10} \\ \hline \end{darray} $$

Combine like terms:

$$ x^7 + x^6 + 2x^3-x^5 + 2x^2-5x^4-2x-10 = \\ x^7+x^6-x^5-5x^4+2x^3+2x^2-2x-10 $$

Polynomial Operations Calculator