Multiply polynomial $ \, \color{blue}{ x^2+2x+1 } \, $ by $ \, \color{orangered}{ x^2-2x+1 }$.

$$ \left( \color{blue}{x^2+2x+1} \right) \cdot \left( \color{orangered}{x^2-2x+1} \right) = x^4-2x^2+1 $$

We can multiply polynomials by using a GRID METHOD

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

$$ \begin{darray}{|c|c|c|c|}\hline & \color{blue}{x^2} & \color{blue}{2x} & \color{blue}{1} \\ \hline \color{blue}{x^2} & & & \\ \hline \color{blue}{-2x} & & & \\ \hline \color{blue}{1} & & & \\ \hline \end{darray} $$

Fill in each empty box by multiplying the intersecting terms.

$$ \begin{darray}{|c|c|c|c|}\hline & \color{blue}{x^2} & \color{blue}{2x} & \color{blue}{1} \\ \hline \color{blue}{x^2} & \color{orangered}{x^4} & \color{orangered}{2x^3} & \color{orangered}{x^2} \\ \hline \color{blue}{-2x} & \color{orangered}{-2x^3} & \color{orangered}{-4x^2} & \color{orangered}{-2x} \\ \hline \color{blue}{1} & \color{orangered}{x^2} & \color{orangered}{2x} & \color{orangered}{1} \\ \hline \end{darray} $$

Combine like terms:

$$ x^4 + 2x^3-2x^3 + x^2-4x^2 + x^2-2x + 2x + 1 = \\ x^4-2x^2+1 $$

Polynomial Operations Calculator