[Math] How to find the square root of a polynomial

Question

Here, by the word polynomial, I am referring to those polynomials, who have a square root in simple algebraic expressions.
But how to get the intuition, on how to expand the polynomial, so that we can factor it, and then find its square root? Or is there any other method also?
Like $\sqrt{\frac{x^2}4+\frac1{x^2}-\frac1x+\frac x2-\frac34}$,
At first look, we get instinct to first take LCM of all and add them. But after that, the numerator becomes $x^4+2x^3-3x^2-4x+4$. Now, (I don't want the answer) how to get that instinct on factoring this polynomial? Or is there a method?
Remark: Is there any faster, non-rigorous way also, of finding the square root?

Answer 1

Hint: Let $t=\dfrac x2-\dfrac1x$ .