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: 
Yes.

In response to your edit: You squared both sides of the equation to correct your post. However, this doesn't make a difference, it's still valid. Since you squared both sides of the equation (applied the same operation), you didn't effectively change anything.

The solution is therefore:

1. take the square root
2. multiply both sides by $(x^{2}-4)$ and divide both sides by $5x$.
3. square both sides ( $1^2=1$ )
Yes, that operation is legal. It just comes from multiplying $\frac{(4-x^2)}{5x}$ on both sides.