Let's find the coefficient of \(x^4\) in the product

$$
\left(1-2 x+4 x^{2}-8 x^{3}+16 x^{4}\right) \cdot\left(1+2 x+4 x^{2}+8 x^{3}+16 x^{4}\right)
$$
By multiplying the appropriate terms, we find that the sought coefficient is 16.