We can multiply polynomials by using a GRID METHOD

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

$$ \begin{darray}{|c|c|c|c|}\hline & \color{blue}{5x^6} & \color{blue}{-3x^4} & \color{blue}{-7} \\ \hline \color{blue}{x^4} & & & \\ \hline \color{blue}{-8x^3} & & & \\ \hline \color{blue}{-2x} & & & \\ \hline \end{darray} $$

Fill in each empty box by multiplying the intersecting terms.

$$ \begin{darray}{|c|c|c|c|}\hline & \color{blue}{5x^6} & \color{blue}{-3x^4} & \color{blue}{-7} \\ \hline \color{blue}{x^4} & \color{orangered}{5x^{10}} & \color{orangered}{-3x^8} & \color{orangered}{-7x^4} \\ \hline \color{blue}{-8x^3} & \color{orangered}{-40x^9} & \color{orangered}{24x^7} & \color{orangered}{56x^3} \\ \hline \color{blue}{-2x} & \color{orangered}{-10x^7} & \color{orangered}{6x^5} & \color{orangered}{14x} \\ \hline \end{darray} $$

Combine like terms:

$$ 5x^{10}-3x^8-40x^9-7x^4 + 24x^7-10x^7 + 56x^3 + 6x^5 + 14x = \\ 5x^{10}-40x^9-3x^8+14x^7+6x^5-7x^4+56x^3+14x $$

Polynomial Operations Calculator