Find all solutions to the equation. cos2x + 2 cos x + 1 = 0

If   cos^2x + 2 cos x + 1 = 0, then we can model this by p^2 + 2p + 1 = 0.
This factors into (p+1)^2.  Set this = 0 and solve for p:  p=-1 and p=-1 (2 real, equal roots).

But p represents cos x.  Thus, cos x = -1.  On the interval [0, 2pi], cos x = -1 for only one x value:  pi.  Thus, x = pi.  Note that x = pi+ 2pi = 3pi is another solution, since the cosine function is periodic and thus repeats itself every 2pi radians.