Solve the equation (5y+1)^2 + 5(5y+1) + 6 = 0.

Do not expand first.

I solved the equation up to 25y^2 + 35y + 12 = 0 and not sure how to proceed from here on.

Do not expand first.

So, let u = 5y+1.
I'm sure you can solve

u^2 + 5u + 6 = 0
(u+3)(u+2) = 0
u = -3 or -2

That means you now need to solve
5y+1 = -3
or
5y+1 = -2

No problem now, eh?