Troubles with a proposed solution path in a 'finding directional derivates' exercise (multivariable calculus)

Hello,

I have troubles understanding how to arrive at certain steps in the proposed solution path.

Here is the question:

And here is the step I have troubles comprehending:

Can you please explain why the third term is 2h^2/4 while the last term is sqrt(2)h^2/4?

And how do get from there to here:

Kind regards

Kosephah

P.S. a formula editor would be nice to have here ;)