I haven't done implicit differentiation in a while, so I might be missing something ... but I managed to get down to the next-to-last line in your book's solution. How they managed to simplify that final numerator to -8, that I don't know.
See the attached pdf.
A suggestion. Maybe your notation is confusing you. Remember that, though the first derivative can written dy/dx, the second derivative is NOT "(dy/dx)^2" and is not written that way. I suggest you write y' for the first derivative and y'' for the second (so you have to remember that you're differentiating with respect to x).
In the pdf, since I'd figured out y', I substituted for that in the equation for y''. I'm not sure I follow your presentation, so I don't know whether you did that or not.
I hope this helps.