Dealing with Cubes...

That's a good step, but the problem with your reasoning is that -7 by itself doesn't have a^3 + 7 in the denominator, so you would have to convert -7 to (-7(a^3 + 7))/(a^3 + 7) before solving.

Rewrite as  :

1/ (a^3 + 7)  +  a^3 / (a^3 + 7)   =   7

(1 + a^3 ) /  (a^3 + 7)   =  7

Multiply both sides  by a^3 + 7

1 + a^3   = 7 ( a^3 + 7)

1 + a^3  = 7a^3  + 49

1 - 49  = 7a^3  - a^3

-48  =  6a^3

-8  = a^3       take the cube root

-2  = a    ( only real solution)