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

$$ \left( \color{blue}{2x^2-x+4} \right) \cdot \left( \color{orangered}{3x^2+x+4} \right) = 6x^4-x^3+19x^2+16 $$

We can multiply polynomials by using a GRID METHOD

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

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

Fill in each empty box by multiplying the intersecting terms.

$$ \begin{darray}{|c|c|c|c|}\hline & \color{blue}{2x^2} & \color{blue}{-x} & \color{blue}{4} \\ \hline \color{blue}{3x^2} & \color{orangered}{6x^4} & \color{orangered}{-3x^3} & \color{orangered}{12x^2} \\ \hline \color{blue}{x} & \color{orangered}{2x^3} & \color{orangered}{-x^2} & \color{orangered}{4x} \\ \hline \color{blue}{4} & \color{orangered}{8x^2} & \color{orangered}{-4x} & \color{orangered}{16} \\ \hline \end{darray} $$

Combine like terms:

$$ 6x^4-3x^3 + 2x^3 + 12x^2-x^2 + 8x^2 + 4x-4x + 16 = \\ 6x^4-x^3+19x^2+16 $$

Polynomial Operations Calculator