> Hmm, let's see now...
>
> φ = (1 + sqrt(5)) / 2
>
> φ + 1 = (1 + sqrt(5)) / 2 + 1 = (3 + sqrt(5)) / 2
>
> φ^2 = (1 + sqrt(5))^2 / 4 = (1 + 2 sqrt(5) + 5) / 4 = 1/4 + sqrt(5)/2 + 
> 5/4
>
> Nope, sorry, I'm not seeing it.

You just need to do a couple more steps:

1/4 + sqrt(5)/2 + 5/4
=
6/4 + sqrt(5)/2
=
3/2 + sqrt(5)/2
=
(3+sqrt(5))/2

:-)

Or you can just solve x+1=x^2 your favourite way.

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