Find the square root of the polynomial My question is:
Find the square root of the polynomial-
$$\frac{x^2}{y^2} + \frac{y^2}{x^2} - 2\left(\frac{x}y + \frac{y}x\right) + 3$$

Let $t=\dfrac xy +\dfrac yx$.
Then $t^2=\dfrac {x^2}{y^2} +\dfrac{y^2}{x^2}+2$.
Your "polynomial" becomes finally an actual polynomial:
$$t^2-2-2t+3=t^2-2t+1=(t-1)^2=\left(\frac xy+\frac yx -1\right)^2.$$
So, the two square roots are $$\pm \left(\frac xy+\frac yx -1\right).$$
