Answer: \(1+4e^{2x}+6e^{4x}+4e^{6x}+e^{8x}\).

The key point is to identify the required binomial coefficients and simplify the powers carefully.

Use the binomial expansion

\((a+b)^4=a^4+4a^3b+6a^2b^2+4ab^3+b^4.\)

Here \(a=1\) and \(b=e^{2x}\), so

\((1+e^{2x})^4 =1+4e^{2x}+6(e^{2x})^2+4(e^{2x})^3+(e^{2x})^4.\)

Therefore

\((1+e^{2x})^4 =1+4e^{2x}+6e^{4x}+4e^{6x}+e^{8x}.\)