Computational complexity of random recursive integer factoring algorithm

I am studying the probabilities PN to find p and q at the j level of the recursion.

I have done many trials to implment the central idea to represent N = x² + bx + c . Solving this quadratic means look for the several values of the random variable x. that satisfy a set of condition in delta and the factors on N . As I could understand and prove c < sqrt(N) .
If delta is a perfect square then b and sqrt(delta ) share their parity and give x_1 and x_2 integers. the probability to extract x such that delta(x) is a perfect square is the point of my question . thank you

Function Factorize(N):
    // Base case: small N
    If N <= 3:
        Return N 

    // 1. Select x in the range [1, floor(sqrt(N))]
    x = RandomInteger(1, floor(sqrt(N)))

    // 2. Compute coefficients based on N = x^2 + bx + c
    // b = floor((N - x^2) / x)
    // c = N - x^2 - (b * x)
    b = floor((N - x^2) / x)
    c = N - x^2 - (b * x)

    // 3. Solve the quadratic congruence delta = b^2 - 4c
    delta = b^2 - (4 * c)

    // 4. If delta is a perfect square, attempt to extract factors
    If delta >= 0 AND IsPerfectSquare(delta):
        sqrt_delta = sqrt(delta)
        x1 = (-b - sqrt_delta) / 2
        x2 = (-b + sqrt_delta) / 2

        p = x + abs(x1)
        q = x + abs(x2)

        If (p * q) == N:
            Return (p, q)

    // 5. Recursive step: descend into the residue c
    Return Factorize(c)