Assuming everything you've done so far is correct:
Solve the quadratic.
I already know my first derivative to be:
So I solved for my second derivative and got:
I'm looking for where is concave up and concave down, and I figured that inflection points were a good way to find that. So, I set my equation equal to zero:
I am completely stuck here. How do I solve for now?
The sign of the second derivative will indicate which direction the function opens: positive second derivative means concave up, negative second derivative means concave down. So those intervals on which the function has a positive second derivative are the intervals in which the function is concave up, vice versa for points where the function has a negative second derivative.