Question: Is this operation allowed?

Going from this: 
$\left ( \frac{x^{2}+6}{x^{2}-4} \right )^{2}= \left ( \frac{5x}{4-x^{2}} \right )^{2}$

To this: $\left ( \frac{\left (x^{2}+6 \right )\left ( 4-x^{2} \right )}{\left (x^{2}-4 \right )5x} \right )^{2}= 1$

Answer: 
Assuming $x \neq 0$, then yes. Otherwise you're dividing by zero.

Hope that helps,
Yes, that operation is legal. It just comes from multiplying $\frac{(4-x^2)}{5x}$ on both sides.