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

$$ \left( \color{blue}{x^2-4x+2} \right) \cdot \left( \color{orangered}{x^2+2x+3} \right) = x^4-2x^3-3x^2-8x+6 $$

We can multiply polynomials by using a GRID METHOD

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

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

Fill in each empty box by multiplying the intersecting terms.

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

Combine like terms:

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

Polynomial Operations Calculator