"Prove 0=3" .Can you support me where is the mistakes?

"Prove 0=3".

Where is mistakes in false proof.

Can you tell me ?

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Mistakes in false proof:

Start with x as a solution to:
Supposed,

x2 + x + 1 = 0

Since x ≠ 0, we can divide both sides by x to get:

x + 1 + 1/x = 0

From the original equation we also have x + 1 = –x2, so we substitute into the equation above to get:

–x2 + 1/x = 0

1/x = x2

1 = x3

x = 1

So we have x = 1, so let’s substitute that back into the original equationx2 + x + 1 = 0:

12 + 1 + 1 = 0

3 = 0

But this conclusion is absurd! And yet every step seemed correct. So where was the mistake?
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.keep continue for  where is mistakes.
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The mistake is...

The first step is fine:

x2 + x + 1 = 0

Then dividing by x is also fine.

x + 1 + 1/x = 0

Both of the above equations have 2 solutions, which are -1/2 + i(√3)/2 and -1/2 – i(√3)/2.

The mistake in the false proof originates in the very next step. We then substitute x + 1 = x2 to get:

–x2 + 1/x = 0

Now this equation actually has 3 solutions. It has the same two complex solutions, but it also has another extraneous solution of x = 1.

As x = 1 is a new solution at this step, we cannot then substitute it back into the original equation.

It’s a pretty sneaky trick in this problem. But the lesson is that not all steps in a proof are reversible, so you have to check for extraneous solutions, or else you could end up with nonsensical results like 3 = 0.

You can also catch me on  given below link.

Thanks for reading this problem.