I'm making a Hexblade warlock with Polearm Master, and I'm thinking about getting a level of fighter for the Dueling Fighting Style.
I have some questions about how these interact with the Eldritch Smite warlock invocation:
The Eldritch Smite warlock invocation (XGtE, p. 56) reads:
Once per turn when you hit a creature with your pact weapon, you can expend a warlock spell slot to deal an extra 1d8 force damage to the target, plus another 1d8 per level of the spell slot, and you can knock the target prone if it is Huge or smaller.
This is an extra die or dice that is/are added to your damage roll, the same with Hex:
Until the spell ends, you deal an extra 1d6 necrotic damage... PHB pg. 251
"Damage rolls" are specified on pg. 196 of the PHB:
Each weapon, spell, and harmful monster ability specifies the damage it deals. You roll the damage die or dice, add any modifiers, and apply the damage to your target.
Since each ability specifically states that your "damage roll" and not "damage die/dice" are getting the boost, this indicates the following:
It does not add an extra +2 to each die that is rolled, just to the overall result of all dice rolled for damage. This also applies to Hexblade's Curse:
You gain a bonus to damage rolls against the cursed target. The bonus equals your proficiency bonus. -XGtE pg. 55
They can stack though, with both being applied at the end of all damage rolled.
Dueling in the PHB on pg. 72 reads as such (emphasis mine):
When you are wielding a melee weapon in one hand and no other weapons, you gain a +2 bonus to damage rolls with that weapon.
Also of note is that if you are wielding the Polearm two-handed, it does not add the extra damage, as it is explicitly stated it must be in one hand.
It should also be kept in mind that the Eldritch Smite invocation is boosting the damage of an attack you are already making, so it is not Eldritch Smite that is getting an additional damage die by Hex but the damage roll while using Eldritch Smite.
The end result of all of the above being used at once, assuming you are using a Quarterstaff as your pact weapon since that is one of only two polearms (the other being a spear with the same damage die and different damage type) that can be used one handed, would be on a hit your damage roll consists of the following: 1d6 bludgeoning (Quarterstaff), 1d6 necrotic (Hex), 2d8 force (Eldritch Smite [assuming level 1 slot]), +2 (dueling style), +2 (Hexblade's Curse adding proficiency bonus assuming low level in both classes), +3 (an assumed value for your Strength modifier)
