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I'm trying to properly understand the Hellfire Warlock's hellfire blast.

Hellfire Blast (Sp): Whenever you use your eldritch blast ability, you can change your eldritch blast into a hellfire blast. A hellfire blast deals your normal eldritch blast damage plus an extra 2d6 points of damage per class level. If your blast hits multiple targets (for example, the eldritch chain or eldritch cone blast shape invocations), each target takes the extra damage. This damage is not fire damage. Hellfire burns hotter than any normal fire, as described in the sidebar on page 119.

Each time you use this ability, you take 1 point of Constitution damage. Because the diabolical forces behind the power of hellfire demand part of your essence in exchange for this granted power, if you do not have a Constitution score or are somehow immune to Constitution damage, you cannot use this ability.

1. Is all the damage from a hellfire blast considered hellfire or only the extra damage?

Eldritch Essence Invocations

Some of a warlock's invocations, such as frightful blast, modify the damage or other effects of the warlock's eldritch blast. These are called eldritch essence invocations. Unless noted otherwise, eldritch blasts modified by eldritch essence invocations deal damage normally in addition to imparting the effects described in the invocation description.

A warlock can apply only one eldritch essence invocation to a single eldritch blast, choosing from any of the eldritch essence invocations that he knows. When a warlock applies an eldritch essence invocation to his eldritch blast, the spell level equivalent of the modified blast is equal to the spell-level of the eldritch blast or of the eldritch essence invocation, whichever is higher. If a warlock targets a creature with an eldritch essence blast that has immunity to the invocation's effect, it still takes the damage from the blast normally (provided it isn't also immune to the eldritch blast).

A warlock can apply an eldritch essence invocation and a blast shape invocation (see below) to the same blast. When a warlock uses both kinds of invocations to alter an eldritch blast, the spell level equivalent is equal to the spell level of the eldritch blast, the level of the eldritch essence invocation, or the level of the blast shape invocation, whichever is higher. (Complete Arcane, pp. 8-9)

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Vitriolic Blast

This eldritch essence invocation allows you to change your eldritch blast into a vitriolic blast. A vitriolic blast deals acid damage, and it is formed from conjured acid, making it different from other eldritch essences because it ignores spell resistance. Creatures struck by a vitriolic blast automatically take an extra 2d6 points of acid damage on following rounds. This acid damage persists for 1 round per five class levels you have. For example, a 15th-level warlock deals 2d6 points of acid damage per round for 3 rounds after the initial vitriolic blast attack. (Complete Arcane, p. 136)

2. Can a hellfire blast also be a vitriolic blast? If it can, is it hellfire or acid? If it's hellfire does it still ignore spell resistance and do an extra 2d6 points of damage on following rounds"?

Here is an interesting discussion on the matter. It ends on post #36.

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  1. It is not strictly unambiguous, but “this damage” most likely—almost certainly, really—refers to the bonus damage added by hellfire blast, and not to the total damage of the eldritch blast it modifies.

  2. Despite modifying eldritch blast in a manner very similar to eldritch essences, Fiendish Codex II fails to actually call it one. This is probably an oversight, and even if it’s not, we’re still left with insufficient details to figure out how things should actually work when you use hellfire blast on a vitriolic blast. Quoting myself from another answer,

    Anyway, it’s not immediately clear whether you can apply hellfire blast and vitriolic blast at the same time, or what happens if you do. Certainly, hellfire blast sounds similar to an eldritch essence, but it never actually says it is one or that you cannot also use one on the hellfire blast. But even if you do, can vitriolic blast turn hellfire into acid? Does it still deal the extra damage if you do? Officially, the answer is “yes,” because nothing nixes that combination and ordering, but it’s worth thinking about.

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I believe it can be reasonably inferred that hellfire blast is an eldritch essence invocation. Eldritch essences change names from "eldritch blast" to "something blast" while shape invocations change blast names from "eldritch blast" to "eldritch something." Also, blast shape invocations do not, without any exception, change the damage dice or type -- that is solely the domain of eldritch essence invocations. The hellfire blast adds damage and changes damage types. As such, a hellfire blast could be combined with a blast shape invocation, such as eldritch line or eldritch chain, but could not be combined with another eldritch essence invocation. For example, one could not combine it with vitriolic blast. Also, the text of hellfire blast gives specific examples of it being combined with blast shape invocations. Blast shape invocations can only be added to an eldritch blast or an eldritch blast modified by an eldritch essence invocation.

As KRyan points out, though, it is never explicitly referred to as an eldritch essence invocation. My table has ruled, though, that its name and rules very strongly suggest that it is one and should follow those rules accordingly.

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