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

$$ \left( \color{blue}{-r^2+3r+4} \right) \cdot \left( \color{orangered}{4r-3} \right) = -4r^3+15r^2+7r-12 $$

We can multiply polynomials by using a GRID METHOD

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

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

Fill in each empty box by multiplying the intersecting terms.

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

Combine like terms:

$$ -4r^3 + 12r^2 + 3r^2 + 16r-9r-12 = \\ -4r^3+15r^2+7r-12 $$

Polynomial Operations Calculator