*The following is touched on in some answers and comments, but I feel is worthy of a separate answer.*

The primary purpose of assignments and exercises like this is *not* (simply) to get you to *produce the correct answer* but to demonstrate to your instructor that you both *understand the concepts involved* and can *convey your understanding* of those concepts.

Starting with a false statement you can "prove" anything. Starting with a *potentially* false statement (or, at least, one whose truth is unknown) will lead to an erroneous proof *unless* you are able to say (and *do* say) that each step (ultimately leading to a "known true" statement) is an example of *if-and-only-if* or *implies-and-is-implied-by* ($\iff$).

Thus, in your answer, even though – as you say – "*Dividing by 2, adding and factoring are pretty obvious right?*" because you have not *explicitly stated* that each step is reversible, your instructor does not know whether:

You **are** aware of the dangers of a "proof from fallacy" but have not bothered to make this clear, or:

You are blissfully **un**aware of those dangers, and – while you got lucky this time – you might use $-5=5 \Rightarrow 25=25$ next time and not notice the proof is invalid.

In a similar vein, when I was at school we were always told to show our full working, because only then could the teachers determine whether we *understood* what were doing or not. There were several outcomes:

*Incorrect answer; no working*: Zero marks.

*Correct answer; no working*: Limited number of marks (probably no more than half the total).

*Incorrect answer; workings shown*: If the workings demonstrated the correct method, but included a simple mistake, you were likely to get reasonable marks (possibly three-quarters).

*Correct answer; workings shown*: Assuming the correct method was used, full marks.

To answer your specific questions:

- Should I have received marks for this question?

_{(original question; as originally answered)}

In this case, I'd be tempted to agree with your instructor. Having chosen a "non-standard" layout, and without explicitly indicating you are aware of the dangers of a "proof from fallacy", while a little harsh, I don't think zero is *overly* harsh. The fact that it has stuck with you sufficiently to come here and ask about it is evidence that you *won't make the same mistake again* which is really the main goal of assignments like this.

- Is my proof equally valid?

_{(after question edited by other than OP, although I believe the original question is more relevant)}

Other answers directly tackle the "correctness" of the OP's proof and I don't really have anything to add (other than a personal view that the assumed *if-and-only-if* nature of each step should have been made more explicit).

The point of *my* answer is that this question is not what someone in the OP's position should be asking (and, to be fair, the OP *didn't* actually ask this question). What is important is *not* whether the proof is correct, but whether the student has demonstrated their understanding of the topic.

So: the proof *may* be correct, but it's still a *bad answer* to the exercise.

- Do real mathematicians all write one way or the other when writing in a paper?

I am not familiar enough with the writing of academic mathematical papers to be certain, but I would *expect* (hope?) that were someone to present a proof in the form you did, they would be explicit in their use of $\iff$ between stages (or, more probably, would have used the "conventional" layout).

_{(An addendum after responding to the edited question.)}

The premise of my answer is that tests/exercises like this are predominately about a student demonstrating that they understand the topic at hand. When I was at school in the UK, studying for what then were O Levels, this was drummed into us constantly: *getting the right answer is less important than showing how you got it*.

You may possibly have a legitimate cause for complaint if your instructors have not sufficiently emphasised this point in the past.

If you feel that this is the case, I'd recommend a conciliatory (rather than confrontational) approach. Explain to your instructor that you now understand *why* they felt unable to give you marks for this question – i.e. because your approach leaves room for not understanding the problems of *proof by fallacy* — but that you feel this need to *demonstrate such understanding* hadn't been sufficiently "*driven home*" in the past, and could they (the instructor) focus more on this aspect in the future.