Proof: $n^2 - 2$ is not divisible by 4

Your proof is perfect.

We can shorten your proof by for example going from $4x^2=4g+2$ (in case 1) to saying "The left-hand side has remainder $0$ after division by $4$, yet the right side has remainder $2$; this is impossible" (basically, looking at the expression $\mod 4$ instead of dividing by $2$ and looking $\mod 2$).

Also, the second case was trivially impossible, since $n^2=4g+2$ has no solutions if $n$ is odd (since then $n^2$ is odd, but $4g+2$ is even).

Depending on the context (what you know, what you can use, etc), steps like $x^2+x=j$ with the remark that integers are closed under addition and multiplication are mostly considered so trivial that it's not worth mentioning. I repeat however that this is completely dependent on context, and if you want to make sure your audience is aware of these facts and/or steps, you should mention them. More detailed explanations with steps rarely hurt the proof.

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The proof can be done a lot quicker however (without contradiction) by looking $\mod 4$. It is quite easy to prove that squares are either $0$ or $1\mod 4$, so $n^2-2$ is either $-2$ or $-1\mod 4$, and thus, $n^2-2$ cannot be divisible by $4$.

For odd $n$, $n^2-2$ is odd.

For even $n$, $n^2$ is divisible by $4$, so that $n^2-2$ is not.

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By contradiction:

Let $n$ be odd. Then $n^2-2$ is both odd and a multiple of four.

Let $n$ be even. Then $n^2-2$ and $n^2$ are both multiples of $4$, so that $2$ is a multiple of $4$.