Query on HCF of polynomials

In the post https://www.mbatious.com/topic/71/hcflcmconceptsbygauravsharma there was a question:
What is the HCF of x^2  5x + 6 and x^2 7x +12 and the solution was:
x^2  5x + 6 = (x2)(x3)
x^2  7x +12 = (x3)(x4)
HCF of [(x2)(x3)] and [(x3)(x4)] cannot be determined as it depends upon the value of xMy question is that why is HCF not (x3) (in schools this is what we were taught for HCF of polynomials). Please shed some light.

there might be common factor in x2 and x4 therefore unless x2 and x4 are coprimes, x3 cannot be the HCF

@namanjain0 Can you please explain the same using following question:
find hcf of x^2 + 7x + 6 and x^2 − 5x − 6.
here the answer should be x + 1 (factors are (x + 1)(x + 6) and (x + 1)(x − 6))
Will the concept of checking for coprimes come into play here too ?

@test1234
If we put x = 1,
x^2 + 7x + 6 = 14 and x^2 − 5x − 6 = 10
Is HCF(14, 10) = 1 + 1 = 2 ?