X being an angle; solve

6 cos^2 x + 5 cos x = -1   add 1 to both sides

6 cos^2 x + 5 cos x  + 1 = 0         factor

(3cosx +1) (2cosx + 1)  = 0

Set each factor to 0

3cosx + 1  =  0                                           2cosx + 1 = 0

3cosx  = -1                                                  2cosx = -1

cosx = -1/3                                                   cosx  = -1/2

arccos (-1/3)  = about109.5° + n360°          arcos (-1/2)  =   120°  + n360°

And at  about     250.5” + n360°                and                     240 °  +  n360°   

Where n is an integer

Here's the graph, Shades :   https://www.desmos.com/calculator/knlolptzrg