Quote:
(1) Not useful for QS
(2) Not really applicable to QS.

We have x  N = (ax + b)^2  N = a *(ax^2 + 2bx + c)  b^2  N = a*c
and we will look for smooth a * (ax^2 + 2bx + c) .
If p is a prime not in the factor base (and n has a quadratic residue mod p) and
ax^2 + 2bx + c != y^2 mod p, then a (ax^2 + 2bx + c) can not be smooth.
So the idea can be used for reducing the possible x in the QS.