Multiply polynomial $ \, \color{blue}{ 4r-5 } \, $ by $ \, \color{orangered}{ 3r^2+9r+4 }$.

$$ \left( \color{blue}{4r-5} \right) \cdot \left( \color{orangered}{3r^2+9r+4} \right) = 12r^3+21r^2-29r-20 $$

We can multiply polynomials by using a GRID METHOD

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

$$ \begin{darray}{|c|c|c|c|}\hline & \color{blue}{3r^2} & \color{blue}{9r} & \color{blue}{4} \\ \hline \color{blue}{4r} & & & \\ \hline \color{blue}{-5} & & & \\ \hline \end{darray} $$

Fill in each empty box by multiplying the intersecting terms.

$$ \begin{darray}{|c|c|c|c|}\hline & \color{blue}{3r^2} & \color{blue}{9r} & \color{blue}{4} \\ \hline \color{blue}{4r} & \color{orangered}{12r^3} & \color{orangered}{36r^2} & \color{orangered}{16r} \\ \hline \color{blue}{-5} & \color{orangered}{-15r^2} & \color{orangered}{-45r} & \color{orangered}{-20} \\ \hline \end{darray} $$

Combine like terms:

$$ 12r^3 + 36r^2-15r^2 + 16r-45r-20 = \\ 12r^3+21r^2-29r-20 $$

Polynomial Operations Calculator