Let's first look at the quadratic function in terms or cosine.
cos2x + 3cosx + 2 = 0
In order to solve for cosx we need to factor the quadratic. To do so it may be easier to rewrite in terms of x instead of cosx.
x2 + 3x + 2 = 0
Factoring this results in:
(x+2)(x+1) = 0
Now we can substitute the cosx back in for the x
(cosx + 2)(cos x +1) = 0
This results in 2 solutions of cos x. cosx = -2 and cosx = -1
cosx is undefined for values greater than 1 and less than -1 (Think of the graph of cosine... it only goes up to 1 and only goes down to -1).
So, cosx = -2 has no solution.
cos x = -1 .... where does cosx = -1 or what angle measures give a value of -1 to cosx. On the interval [0,2π] this happens only at π.
So if we are asked to solve on the interval [0,2π], the final answer would be x = π.
But, if does not give us an interval they are likely wanting us to give a general solution which accounts for how the cosine function repeats from -∞ to ∞. In this case we need to add the period of the cosine function multiplied by n. As a refresher the period is the distance on the x-axis over which the cosine function takes to repeat itself. The period of cosx = 2π.
Therefore, the general solution is:
x = π + 2nπ FINAL ANSWER
Hope this explanation helps.