I know how to graph them but it says I have to solve the system of equations...How to I solve that?

y  =  4 cos (x) + 1

y  = -cos (x) - 4              set the y's equal.....and we have

4cos (x) + 1  = -cos(x) - 4         add  cos (x), 4 to both sides

5cos (x) + 5  =  0        subtract  5 from both sides

5 cos (x)  = - 5              divide both sides by 5

cos (x)  =  -1

And this happens when x  =  pi    and x  = -pi

The intersection point(s) are

4cos (pi) + 1  =  -4 + 1  =  -3   ⇒     (pi, -3)         and

4cos(-pi) + 1  =  -4 + 1  =  - 3  ⇒  (-pi, -3)

See the graph here, GM : https://www.desmos.com/calculator/j9yashts7t

This is called 'solving by the graphical method'..... when you graph the two (or more ) equations......The points where they intersect is the solution set.