For the solutions of 4tantheta = 5tantheta + sqroot3, can both these answers be acceptable?

Question

120 + 180n
-60 + 180n

Answer 1

-tanØ = √3
tanØ = -√3 , so Ø is in II or IV

first primary answer = 120°
2nd primary answer = 300° , notice the difference between them is 180° and the tangent curve has a period of 180° so we can get away with a single
general solution:
(120 + 180n)° , where n is an integer

(notice when n = 1, we get my 2nd answer,
if n = -1 , we get your answer of -60°)

both of your answers would of course be acceptable, but as noted, the second one is redundant.