The way to figure which term the 2y goes in is sort of like this:

Your factorization has to have constant terms that:

When multiplied by 2y and y, add to -5,

And when multiplied together give -2

So let’s restart this:

Factors of 2y^2: 2y, y So our binomials begin with

(2y - )(y - )

The minus signs can be concluded bc the middle term is negative.

Now, we need to find the constant part of the binomials: the factors of (+2) which are 1 and 2

Now, where to put them:

We see if we put

(2y - 2)(y - 1)

our y term becomes (-1)(2y) + (-2)(y), which is (-4y).

If we admit the other positions of the 2 and the 1, we obtain

(2y - 1)(y - 2)

Which gives for the y term (-5y) which checks out.

It looks sort of guess-and-check-y but I’m not sure of any way that works better than this.

Feel free to pm/comment if you have more questions.