Any shortcuts to factoring Trinomials when looking for numbers that when multiplied together equal y and when added together equal z?

So, how I understand (simple) Trinomials is essentially;

x^2 + yx + z = (x + a)(x+b)

(Where yx = ax + bx & zx = ab)

Non-simple trinomals are a little different;

(A)x^2 + yx + z

(where A > than 1)

but the Problem I am having with them, Is essentially the same, and comes down to this part;

(Where yx = ax + bx & zx = ab)

I.E finding what combinations, when multiplied together equal y and when added together equal z.

Other than simply guess and check, is there really any want to shortcut this?