Multiply polynomial $ \, \color{blue}{ x^4+5x^3 } \, $ by $ \, \color{orangered}{ 5x^4-4x^3-5x^2 }$.

$$ \left( \color{blue}{x^4+5x^3} \right) \cdot \left( \color{orangered}{5x^4-4x^3-5x^2} \right) = 5x^8+21x^7-25x^6-25x^5 $$

We can multiply polynomials by using a GRID METHOD

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

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

Fill in each empty box by multiplying the intersecting terms.

$$ \begin{darray}{|c|c|c|c|}\hline & \color{blue}{5x^4} & \color{blue}{-4x^3} & \color{blue}{-5x^2} \\ \hline \color{blue}{x^4} & \color{orangered}{5x^8} & \color{orangered}{-4x^7} & \color{orangered}{-5x^6} \\ \hline \color{blue}{5x^3} & \color{orangered}{25x^7} & \color{orangered}{-20x^6} & \color{orangered}{-25x^5} \\ \hline \end{darray} $$

Combine like terms:

$$ 5x^8-4x^7 + 25x^7-5x^6-20x^6-25x^5 = \\ 5x^8+21x^7-25x^6-25x^5 $$

Polynomial Operations Calculator