Is it technically incorrect to write proofs forward?

A question on an assignment was similar to prove: $$2a^2-7ab+2b^2 \geq -3ab.$$ and my proof was: $$2a^2-4ab+2b^2\geq0$$ $$a^2-2ab+b^2\geq0$$ $$(a-b)^2\geq0$$ which is true.

However, my professor marked this as incorrect and the "correct" way to do it was:

Starting from $$(a-b)^2\geq0$$ we have: $$a^2-2ab+b^2\geq0$$ $$2a^2-4ab+2b^2\geq0$$ $$2a^2-7ab+2b^2 \geq -3ab.$$

His point was that if we start with a false statement we can also reduce it to a true statement (like $-5 =5$ we can square for $25 = 25$). I argued however that you can go back and forth between my operations (if and only if, which doesn't work for the $-5 =5$ example). He still didn't give me the marks for it. Which leads me to my questions:

1. Is my proof equally valid?

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

There are two ways to interpret your argument (which hardly counts as a proof if this is exactly what you have written in your assignment).

First Way

You actually meant to write, \begin{align}2a^2-7ab+2b^2 \geq -3ab&\implies 2a^2-4ab+2b^2\geq0\\&\implies a^2-2ab+b^2\ge 0\\&\implies (a-b)^2\ge0\end{align}

Second Way

You actually meant to write, \begin{align}2a^2-7ab+2b^2 \geq -3ab&\iff 2a^2-4ab+2b^2\geq0\\&\iff a^2-2ab+b^2\ge 0\\&\iff (a-b)^2\ge0\end{align}

In the second case your argument is correct but in the first case it is not (as your professor has elaborated via an example).

Now the answer to your questions,

1. If you didn't say explicitly in your assignment in which way your argument is to be interpreted and then if your teacher interprets your argument in the first way, you can't blame him/her for not giving you the corresponding marks of the question simply because it was you who failed to be explicit )and the first way of interpreting your argument indeed shows a misunderstanding of the working of $\implies$). So, if I were in the position of your professor, I would give you no marks.

2. I don't know what you mean by "one way or the other" here. The best way to answer this question will be to read the papers of some real mathematicians. However, when real mathematicians write a paper it is in general clear what they are assuming to be true and how the steps lead to the conclusion (which is not the case in your argument as I have explained above).

Your professor is correct. Whether or not it's clear "what you mean," in my teaching experience this mistake exposes an extremely common misconception about how proofs work. In a proof, you're trying to communicate how a logical conclusion follows from one or more logical premises. A proof is never just a sequence of assertions: it must communicate the logical relationships between those assertions. Whenever we get lazy and just write the assertions, the implicit logical relationship is that each one follows as a logical consequence from the assertions written before it. That's not what you want here, though: the premise needs to be something you know, and the conclusion needs to be what you're trying to show.

So, whenever you're tempted to write one assertion after another, think to yourself: is this assertion logically equivalent to the one before it? Does it imply the one before it? Is it implied by the one before it? The sooner you get in the habit of putting those relationships in writing, the better. As commenters have said, it can be as simple as putting little arrows between your statements that point from logical premise to logical conclusion.

(In "real" mathematical writing, you can write premises before conclusions or conclusions before premises, but it's always clear which is which. It should be that way in your writing, too.)

What you wrote, unless you made the equivalence marks explicit, is:

\begin{align}2a^2-7ab+2b^2 \geq -3ab\implies ...\implies (a-b)^2\ge0\end{align}

In other words you've shown that if the left side is true, then it follows that the right side is too. Knowing that the right side is true doesn't allow you to say anything about the left side. (What you can say, though, is that if the right side is false, then it'll follow that the left side is also false.)

What your professor wrote, by contrast, and assuming no explicit equivalence marks either, is:

\begin{align}(a-b)^2\ge0\implies ...\implies 2a^2-7ab+2b^2 \geq -3ab\end{align}

And thus, because the left side is true, it follows by implication that the right side is too.

1. Should I have received marks for this question?

I don't teach, but FWIW I wouldn't have given you any marks either.

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

It's not about writing it one way or the other, it's about getting the implications in the correct order.

I'm going to emphasize something that I don't see in the other answers. The point of a proof is ultimately to be a persuasive argument to the reader (Richard Lipton has written on this at least once). The first statement should be a self-evidently true fact (or an assumption of the hypothesis). It would be weird for the reader to be in the dark as to the truthfulness of all the statements in the proof until the end of the reading.

I will use as an example a fragment from the Sherlock Holmes story, "Adventure of the Dancing Men". In one scene Holmes astonishes Watson by deducing from a small indentation on his finger that he will not be investing in South African securities. He reasons aloud:

1. You had chalk between your left finger and thumb when you returned from the club last night.
2. You put chalk there when you play billiards, to steady the cue.
3. You never play billiards except with Thurston.
4. You told me, four weeks ago, that Thurston had an option on some South African property which would expire in a month, and which he desired you to share with him.
5. Your check book is locked in my drawer, and you have not asked for the key.
6. You do not propose to invest your money in this manner.

Consider if this series of statements were reversed:

1. You do not propose to invest your money in this manner.
2. Your check book is locked in my drawer, and you have not asked for the key.
3. You told me, four weeks ago, that Thurston had an option on some South African property which would expire in a month, and which he desired you to share with him.
4. You never play billiards except with Thurston.
5. You put chalk there when you play billiards, to steady the cue.
6. You had chalk between your left finger and thumb when you returned from the club last night.

While not entirely incoherent, the reversed sequence seems noticeably harder to follow the connections than the original. This is similar to what you were doing in your proposed proof. While the convention is largely a stylistic one, it is indeed mostly followed by mathematicians -- but moreover, presenting in the forward direction is widely followed by any good writer or speaker, who wants to make their thought as transparent and persuasive as possible, to any reader or listener.

Notice that in the original text, Holmes starts with something that is undeniably verifiable by everyone present (the fact that Watson has an indentation on his finger), and gets him to agree to that first. It is well known psychologically that saying "yes" primes to say another "yes" in sequence (in sales, this is called the Yes-Set Close). While we should use such techniques responsibly, as persuasive writers, we should not fail to use such tools if they are available.