Absolute of a complex number or modulus is defined as

|a+bi| = \sqrt{a^2+b^2}

e^{iθ} = \cosθ +i\sinθ

Let x = |e^{iθ}-2| + |e^{iθ}+2|

Then

x=\sqrt{(\cosθ+2)^2+\sin^2θ} +\sqrt{(\cosθ-2)^2+\sin^2θ}

=\sqrt{5+2\cosθ} +\sqrt{5-2\cosθ}

Squaring both sides of the equation gives

x^2 = 10+2\sqrt{5^2-4\cos^2θ}

which shows x has maximum value when \cosθ=0. Then we have,

x_{max}^2 = 20

x_{max} = 2\sqrt{5}

Therefore, C is the right choice.