Yes, indeed Tito! I accidentally was referring to the second family rather than the first in the last two sentences of my general comment on specialization at degenerate points. Your last two comments have it right.

In general, when you have a family with generic Galois group G and you specialize at a perhaps degenerate point then the Galois group of the specialization is a subquotient of G. For the family from the original post, the degenerate points are 0 and -81/16. This comment by itself does not force the quintic at 0 to be solvable, but a closer analysis at the nature of the degeneration would. On the other hand, at -81/16 the polynomial factors as quintic (sextic)^2. The quintic has Galois group S5 and "pure thought" forces the sextic to be non-generic, with Galois group PGL2(5)=S5.