Multiply polynomial $ \, \color{blue}{ -6b+7 } \, $ by $ \, \color{orangered}{ 2b^2+7b+3 }$.

$$ \left( \color{blue}{-6b+7} \right) \cdot \left( \color{orangered}{2b^2+7b+3} \right) = -12b^3-28b^2+31b+21 $$

We can multiply polynomials by using a GRID METHOD

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

$$ \begin{darray}{|c|c|c|c|}\hline & \color{blue}{2b^2} & \color{blue}{7b} & \color{blue}{3} \\ \hline \color{blue}{-6b} & & & \\ \hline \color{blue}{7} & & & \\ \hline \end{darray} $$

Fill in each empty box by multiplying the intersecting terms.

$$ \begin{darray}{|c|c|c|c|}\hline & \color{blue}{2b^2} & \color{blue}{7b} & \color{blue}{3} \\ \hline \color{blue}{-6b} & \color{orangered}{-12b^3} & \color{orangered}{-42b^2} & \color{orangered}{-18b} \\ \hline \color{blue}{7} & \color{orangered}{14b^2} & \color{orangered}{49b} & \color{orangered}{21} \\ \hline \end{darray} $$

Combine like terms:

$$ -12b^3-42b^2 + 14b^2-18b + 49b + 21 = \\ -12b^3-28b^2+31b+21 $$

Polynomial Operations Calculator