How do I properly use Ricci calculus in this example?

[jdbbou]

Do I substitute $A_\mu + \partial_\mu \lambda$ everywhere $A_\mu$ appears, then expand out? Do I substitute a contravariant form of the substitution for $A^\mu$ as well? (If so, do I use a metric to convert it first?)

I’m new to Ricci calculus; an explanation as to the meaning of raised and lowered indices here would be greatly appreciated. Everything I've found online so far has been (unnecessarily, I feel) complicated.

 

[jdbbou]

This is what I've tried so far. Any suggestions? Was my justification for step (1) correct?

 

[jdbbou]

For anyone with the same issue in future:
http://physics.stackexchange.com/qu...rangian-local-gauge-invariance-correct#268960