Algebra question resubmited

Hello I have a quick algebra question.

If I have the following expression

$\large \frac{1}{x^2+1}$, and I multiply the numerator and the denominator by $(x^2+1)$.

Is there any way I can get $x^4+x^2−1$?

The reason I am asking this is because on a problem I did this expression came up and it confused I will post the link.

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This is the question where the expression occurs math.stackexchange.com/questions/260117/… –  Fernando Martinez Dec 22 '12 at 15:08
Your previous question was closed. This is a literal copy. Why did you think it wouldn't get closed this time? –  akkkk Dec 22 '12 at 15:12
Previous closed question math.stackexchange.com/questions/263482/… –  Henry Dec 22 '12 at 15:29
@DavidMitra: only because of the later edits by Crete and amWhy after my comment, and the question is still not clear to me. –  akkkk Dec 22 '12 at 15:35
In the other question you like you have an equation: $x^2 = \frac{1}{x^2 + 1}$. Clearing the denominator, you have $x^4 + x^2 = 1$ which you just rewrite as $x^4 + x^2 - 1 = 0$... –  Tyler Dec 22 '12 at 15:43

Essentially repeating Tyler Bailey's comment, in the answer to the linked question you faced solving $$x=\dfrac{1}{\sqrt{x^2+1}}.$$

Squaring both sides, noting this might introduce spurious solutions, gave $$x^2=\dfrac{1}{{x^2+1}}.$$

Multiplying both sides by $x^2+1$ (not numerator and denominator), which might lead to spurious solutions of the form $x=\pm i$ but in fact does not here, gave $$x^4 +x^2 =1.$$

Subtracting $1$ from both sides gave $$x^4 +x^2 -1 = 0.$$

This is about changes to an equation not an expression.

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Oh I see now gracias. –  Fernando Martinez Jan 10 at 3:18