s_e= a |+-> + b |-+>
s_n= a |+-> + b |-+> + c |--> + d |++>
a|+-> + b|-+> is a very special-case entanglement. Yes, you'll see that one mentioned in Wikipedia (and quantum cryptography and etc) because it's very simple.
The general form for the states of two photons is exactly as you have listed for s_n. For some coefficient values of a, b, c, d, the photons are "entangled", for some, they are not "entangled". How do you know which is which? It is whether you can factor the polynomial into two separate expressions of the form (x|+> + y|->). If you can do this, the photons are not entangled; if you cannot do it, they are entangled.
Since most sets of (a, b, c, d) represent unfactorable expressions, you would expect almost any expression chosen at random to represent an entangled pair. Clean un-entanglement is the rare exception.
This is true only for pure-state entanglement (which itself is a rare exception ;-).
This even works if the states are known! This is how quantum cryptography works. You put two particles into Bell state with each other, give one to someone else, and then decide later what message you want to send.
I recommend learning the math. It is pretty simple if you already have linear algebra.