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

Notice that the denominators are equal, save for a factor of $-1$:  which has no impact since the fraction is squared. $$\; (4 - x^2)^2 = (-(x^2 - 4))^2 = (x^2 - 4)^2$$
So multiply both sides by $(x^2 - 4)^2$, and you'll cancel both denominators. Then there's no need to divide by $5x$. 
After multiplying both sides by $(x^2 - 4)^2$, we get
$$\left (x^{2}+6\right )^{2}= \left ( 5x \right )^{2}\iff (x^4 + 12x^2 + 36) - 25x^2 = 0 \iff x^4 - 13x^2 + 36 = (x^2 - 9)(x^2 - 4) = (x+3)(x-3)(x+2)(x-2) = 0  $$
But we need to throw out the solutions $x = 2, -2$ because the equation is not defined there.
