Multiply polynomial $ \, \color{blue}{ 3q^2+5q } \, $ by $ \, \color{orangered}{ 6q^2+8q+9 }$.

$$ \left( \color{blue}{3q^2+5q} \right) \cdot \left( \color{orangered}{6q^2+8q+9} \right) = 18q^4+54q^3+67q^2+45q $$

We can multiply polynomials by using a GRID METHOD

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

$$ \begin{darray}{|c|c|c|c|}\hline & \color{blue}{6q^2} & \color{blue}{8q} & \color{blue}{9} \\ \hline \color{blue}{3q^2} & & & \\ \hline \color{blue}{5q} & & & \\ \hline \end{darray} $$

Fill in each empty box by multiplying the intersecting terms.

$$ \begin{darray}{|c|c|c|c|}\hline & \color{blue}{6q^2} & \color{blue}{8q} & \color{blue}{9} \\ \hline \color{blue}{3q^2} & \color{orangered}{18q^4} & \color{orangered}{24q^3} & \color{orangered}{27q^2} \\ \hline \color{blue}{5q} & \color{orangered}{30q^3} & \color{orangered}{40q^2} & \color{orangered}{45q} \\ \hline \end{darray} $$

Combine like terms:

$$ 18q^4 + 24q^3 + 30q^3 + 27q^2 + 40q^2 + 45q = \\ 18q^4+54q^3+67q^2+45q $$

Polynomial Operations Calculator