Multiply polynomial $ \, \color{blue}{ 3x+1 } \, $ by $ \, \color{orangered}{ 3x^2-10x-3 }$.

$$ \left( \color{blue}{3x+1} \right) \cdot \left( \color{orangered}{3x^2-10x-3} \right) = 9x^3-27x^2-19x-3 $$

We can multiply polynomials by using a GRID METHOD

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

$$ \begin{darray}{|c|c|c|c|}\hline & \color{blue}{3x^2} & \color{blue}{-10x} & \color{blue}{-3} \\ \hline \color{blue}{3x} & & & \\ \hline \color{blue}{1} & & & \\ \hline \end{darray} $$

Fill in each empty box by multiplying the intersecting terms.

$$ \begin{darray}{|c|c|c|c|}\hline & \color{blue}{3x^2} & \color{blue}{-10x} & \color{blue}{-3} \\ \hline \color{blue}{3x} & \color{orangered}{9x^3} & \color{orangered}{-30x^2} & \color{orangered}{-9x} \\ \hline \color{blue}{1} & \color{orangered}{3x^2} & \color{orangered}{-10x} & \color{orangered}{-3} \\ \hline \end{darray} $$

Combine like terms:

$$ 9x^3-30x^2 + 3x^2-9x-10x-3 = \\ 9x^3-27x^2-19x-3 $$

Polynomial Operations Calculator