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

$$ \left( \color{blue}{4p^2+p+3} \right) \cdot \left( \color{orangered}{p-3} \right) = 4p^3-11p^2-9 $$

We can multiply polynomials by using a GRID METHOD

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

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

Fill in each empty box by multiplying the intersecting terms.

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

Combine like terms:

$$ 4p^3 + p^2-12p^2 + 3p-3p-9 = \\ 4p^3-11p^2-9 $$

Polynomial Operations Calculator