The number to be factored is $N$.
Steps

Create the factor base. Let it be List $FB = {p1, p2, p3, p4, …}$

Generate a list which needs to be sieved down. Let this be List $L = {x1, x2, x3, x4….}$

The quadratic polynomial we are using is $f(x) = x^2 – N$
It’s trivially proved that if $f(x) pmod p1 equiv 0$, then $f(x + kp1) pmod p1 equiv 0$ 
Starting with $p1$, solve $x^2 – N equiv 0 pmod p1$

If we find a number(root) $x0$ as the solution of the above, then it’s divisible by $p1$ & $(x0+p1, x0+2p1, x0+3p1..)$ are also divisible by $p1$. So we can sieve the List L using this.
I understood up to this much.
However, once we reach this stage, how does one find the first number in List $L$ which is divisible by $p1$? Do we need to go trying $x1, x2, x3$ etc by trial division till we find the first one divisible by $p1$? Once I find it, then other numbers like $x1 + p1, x1 + 2p2$ etc can easily be found without further division.
However, is going through the List $L$ by trial division the way to find the first one or is there a better way which I am missing?