Multiply polynomial $ \, \color{blue}{ 2p-1 } \, $ by $ \, \color{orangered}{ -p^2+4p-1 }$.

$$ \left( \color{blue}{2p-1} \right) \cdot \left( \color{orangered}{-p^2+4p-1} \right) = -2p^3+9p^2-6p+1 $$

We can multiply polynomials by using a GRID METHOD

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

$$ \begin{darray}{|c|c|c|c|}\hline & \color{blue}{-p^2} & \color{blue}{4p} & \color{blue}{-1} \\ \hline \color{blue}{2p} & & & \\ \hline \color{blue}{-1} & & & \\ \hline \end{darray} $$

Fill in each empty box by multiplying the intersecting terms.

$$ \begin{darray}{|c|c|c|c|}\hline & \color{blue}{-p^2} & \color{blue}{4p} & \color{blue}{-1} \\ \hline \color{blue}{2p} & \color{orangered}{-2p^3} & \color{orangered}{8p^2} & \color{orangered}{-2p} \\ \hline \color{blue}{-1} & \color{orangered}{p^2} & \color{orangered}{-4p} & \color{orangered}{1} \\ \hline \end{darray} $$

Combine like terms:

$$ -2p^3 + 8p^2 + p^2-2p-4p + 1 = \\ -2p^3+9p^2-6p+1 $$

Polynomial Operations Calculator