We can multiply polynomials by using a GRID METHOD

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

$$ \begin{darray}{|c|c|c|c|}\hline & \color{blue}{x^2} & \color{blue}{-4x} & \color{blue}{5} \\ \hline \color{blue}{x^3} & & & \\ \hline \color{blue}{-7x^2} & & & \\ \hline \color{blue}{8} & & & \\ \hline \end{darray} $$

Fill in each empty box by multiplying the intersecting terms.

$$ \begin{darray}{|c|c|c|c|}\hline & \color{blue}{x^2} & \color{blue}{-4x} & \color{blue}{5} \\ \hline \color{blue}{x^3} & \color{orangered}{x^5} & \color{orangered}{-4x^4} & \color{orangered}{5x^3} \\ \hline \color{blue}{-7x^2} & \color{orangered}{-7x^4} & \color{orangered}{28x^3} & \color{orangered}{-35x^2} \\ \hline \color{blue}{8} & \color{orangered}{8x^2} & \color{orangered}{-32x} & \color{orangered}{40} \\ \hline \end{darray} $$

Combine like terms:

$$ x^5-4x^4-7x^4 + 5x^3 + 28x^3 + 8x^2-35x^2-32x + 40 = \\ x^5-11x^4+33x^3-27x^2-32x+40 $$

Polynomial Operations Calculator