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2sin^2 x  + 5sin x  - 3    =  0      factor as

(2sin x   - 1) ( sin x   + 3)  =  0

The only  solution(s)  come(s) from setting the  first factor to 0 and solving

So   we  have

2sin x  - 1   = 0

2sin x  =  1

sin x  = 1/2

This happens  at   

pi/6  + n *2pi 

And

(5pi)/6  + n* 2 pi                where n is an integer