Find all real solutions for x in \( 2(2^x- 1) x^2 + (2^{x^2}-2)x = 2^{x+1} -2 . \)

Thanks for your time!

The solutions are x = -2, x = 2, and x = 5.

Could you please explain to me how you got those answers?

Can you confirm whether you intend

\({2^{x^{2}}}={2^{(x^{2})}}\quad or \quad {(2^{x}})^{2}\)

I think if your notation is correct it should be the first one.

That is the one I have been playing with. But so far I am baffled.

Yeah, I mean the former.

Assuming \(2^{x^2} \) means \(2^{(x^2)}\) then:

Thanks!