Multiply polynomial $ \, \color{blue}{ x^2-2x-2 } \, $ by $ \, \color{orangered}{ 4x^2-4x+1 }$.

$$ \left( \color{blue}{x^2-2x-2} \right) \cdot \left( \color{orangered}{4x^2-4x+1} \right) = 4x^4-12x^3+x^2+6x-2 $$

We can multiply polynomials by using a GRID METHOD

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

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

Fill in each empty box by multiplying the intersecting terms.

$$ \begin{darray}{|c|c|c|c|}\hline & \color{blue}{x^2} & \color{blue}{-2x} & \color{blue}{-2} \\ \hline \color{blue}{4x^2} & \color{orangered}{4x^4} & \color{orangered}{-8x^3} & \color{orangered}{-8x^2} \\ \hline \color{blue}{-4x} & \color{orangered}{-4x^3} & \color{orangered}{8x^2} & \color{orangered}{8x} \\ \hline \color{blue}{1} & \color{orangered}{x^2} & \color{orangered}{-2x} & \color{orangered}{-2} \\ \hline \end{darray} $$

Combine like terms:

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

Polynomial Operations Calculator