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

$$ \left( \color{blue}{x^2-3x-4} \right) \cdot \left( \color{orangered}{2x^2+5x+2} \right) = 2x^4-x^3-21x^2-26x-8 $$

We can multiply polynomials by using a GRID METHOD

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

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

Fill in each empty box by multiplying the intersecting terms.

$$ \begin{darray}{|c|c|c|c|}\hline & \color{blue}{x^2} & \color{blue}{-3x} & \color{blue}{-4} \\ \hline \color{blue}{2x^2} & \color{orangered}{2x^4} & \color{orangered}{-6x^3} & \color{orangered}{-8x^2} \\ \hline \color{blue}{5x} & \color{orangered}{5x^3} & \color{orangered}{-15x^2} & \color{orangered}{-20x} \\ \hline \color{blue}{2} & \color{orangered}{2x^2} & \color{orangered}{-6x} & \color{orangered}{-8} \\ \hline \end{darray} $$

Combine like terms:

$$ 2x^4-6x^3 + 5x^3-8x^2-15x^2 + 2x^2-20x-6x-8 = \\ 2x^4-x^3-21x^2-26x-8 $$

Polynomial Operations Calculator