Multiply polynomial $ \, \color{blue}{ 3x^2-7x+5 } \, $ by $ \, \color{orangered}{ x^2-4x }$.

$$ \left( \color{blue}{3x^2-7x+5} \right) \cdot \left( \color{orangered}{x^2-4x} \right) = 3x^4-19x^3+33x^2-20x $$

We can multiply polynomials by using a GRID METHOD

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

$$ \begin{darray}{|c|c|c|c|}\hline & \color{blue}{3x^2} & \color{blue}{-7x} & \color{blue}{5} \\ \hline \color{blue}{x^2} & & & \\ \hline \color{blue}{-4x} & & & \\ \hline \end{darray} $$

Fill in each empty box by multiplying the intersecting terms.

$$ \begin{darray}{|c|c|c|c|}\hline & \color{blue}{3x^2} & \color{blue}{-7x} & \color{blue}{5} \\ \hline \color{blue}{x^2} & \color{orangered}{3x^4} & \color{orangered}{-7x^3} & \color{orangered}{5x^2} \\ \hline \color{blue}{-4x} & \color{orangered}{-12x^3} & \color{orangered}{28x^2} & \color{orangered}{-20x} \\ \hline \end{darray} $$

Combine like terms:

$$ 3x^4-7x^3-12x^3 + 5x^2 + 28x^2-20x = \\ 3x^4-19x^3+33x^2-20x $$

Polynomial Operations Calculator