Understanding the Problem:
To rationalize the given expression \(\frac{2}{2\sqrt{3}+4\sqrt{5}}\), we need to eliminate the radical in the denominator by multiplying both the numerator and denominator by the conjugate of the denominator.

Steps to Rationalize:
- Identify the conjugate of the denominator
- Multiply the numerator and denominator by the conjugate
- Simplify the expression

Identifying the Conjugate:
The conjugate of the denominator \(2\sqrt{3}+4\sqrt{5}\) is \(2\sqrt{3}-4\sqrt{5}\).

Multiplying by the Conjugate:
\[
\frac{2}{2\sqrt{3}+4\sqrt{5}} \times \frac{2\sqrt{3}-4\sqrt{5}}{2\sqrt{3}-4\sqrt{5}}
\]
This results in:
\[
\frac{4\sqrt{3}-8\sqrt{5}}{4(3)-16(5)}
\]
Simplify the expression further:
\[
\frac{4\sqrt{3}-8\sqrt{5}}{12-80}
\]
\[
\frac{4\sqrt{3}-8\sqrt{5}}{-68}
\]
Thus, the rationalized form of the given expression is \(\frac{4\sqrt{3}-8\sqrt{5}}{-68}\).