Multiply polynomial $ \, \color{blue}{ -3n^2+n-1 } \, $ by $ \, \color{orangered}{ 3n+4 }$.

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

We can multiply polynomials by using a GRID METHOD

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

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

Fill in each empty box by multiplying the intersecting terms.

$$ \begin{darray}{|c|c|c|c|}\hline & \color{blue}{-3n^2} & \color{blue}{n} & \color{blue}{-1} \\ \hline \color{blue}{3n} & \color{orangered}{-9n^3} & \color{orangered}{3n^2} & \color{orangered}{-3n} \\ \hline \color{blue}{4} & \color{orangered}{-12n^2} & \color{orangered}{4n} & \color{orangered}{-4} \\ \hline \end{darray} $$

Combine like terms:

$$ -9n^3 + 3n^2-12n^2-3n + 4n-4 = \\ -9n^3-9n^2+n-4 $$

Polynomial Operations Calculator