Homework Help: Applying the Chain Rule (Calculus)?

[deathapult2]

(deleted other question because I forgot to add some stuff)

I'm currently struggling with the chain rule for weeks now, and I'm still not used to this. So far, I've ended up revising the problem four times due to small mistakes. Right now I'm trying to apply the chain rule to this part:

f''(x) = [ -2( x² + 1 )² + 8x ] = numerator
_________(x²+1)³ = denominator

So far, I end up getting:

[ -2(x²+1)[ (x²+1) + 4x ] ]
______(x²+1)³

[ -2(x²+1)[ (x² + 4x + 1) ]
______(x²+1)³

...and I don't know what to do beyond that. In fact, I don't even know if I did this right. Normally, it would be the answer, but I'm trying to find the critical points of this equation to determine the concavity and the points of inflection, so there are more steps I have to take. Any help would be appreciated...

(I'm using the quotient rule, btw. I'm not used to being capable of dealing with the squared part of the problem. (in other words, (x²+1)².)
EDIT: Forgot to mention - will need steps to getting the critical points as well.
EDIT: I know how to do the concavity and such, it's just this part that I have problems with, but yeah, thanks for your help!