We can multiply polynomials by using a GRID METHOD

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

$$ \begin{darray}{|c|c|c|c|c|}\hline & \color{blue}{10x^3} & \color{blue}{3x^2} & \color{blue}{-10x} & \color{blue}{-3} \\ \hline \color{blue}{4x} & & & & \\ \hline \color{blue}{-9} & & & & \\ \hline \end{darray} $$

Fill in each empty box by multiplying the intersecting terms.

$$ \begin{darray}{|c|c|c|c|c|}\hline & \color{blue}{10x^3} & \color{blue}{3x^2} & \color{blue}{-10x} & \color{blue}{-3} \\ \hline \color{blue}{4x} & \color{orangered}{40x^4} & \color{orangered}{12x^3} & \color{orangered}{-40x^2} & \color{orangered}{-12x} \\ \hline \color{blue}{-9} & \color{orangered}{-90x^3} & \color{orangered}{-27x^2} & \color{orangered}{90x} & \color{orangered}{27} \\ \hline \end{darray} $$

Combine like terms:

$$ 40x^4 + 12x^3-90x^3-40x^2-27x^2-12x + 90x + 27 = \\ 40x^4-78x^3-67x^2+78x+27 $$

Polynomial Operations Calculator