Correct Answer: $\sqrt{129}$

Solution : Given: $x^4+\frac{16}{x^4}=15617$
⇒ $(x^2)^2 + (\frac{4}{x^2})^2 + 2\times x^2 \times \frac{4}{x^2} = 15617 + 2\times x^2 \times \frac{4}{x^2}$
⇒ $(x^2+\frac{4}{x^2})^2 = 15625$
⇒ $x^2 + \frac{4}{x^2} = 125$
⇒ $x^2 + (\frac{2}{x})^2 + 2\times x \times \frac{2}{x} = 125 + 2\times x\times \frac{2}{x}$
⇒ $(x+\frac{2}{x})^2 = 129$
⇒ $x+\frac{2}{x} = \sqrt{129}$
Hence, the correct answer is $\sqrt{129}$.