Question 854542
Your original equation makes little sense. The expression *[tex \large \sin^3] is meaningless, and there are two equal signs. I presume your original equation is *[tex \large 6 \sin^3 x - 18 \sin^2 x + 5 \sin x + 15], given the way you factored it (which should be *[tex \large 6 \sin^2 x (\sin x + 3) = 5 \sin x + 15]).


Note that you can divide both sides by *[tex \large \sin x + 3], since *[tex \large \sin x + 3 \neq 0] for real x, obtaining


*[tex \large 6 \sin^2 x = 5]


*[tex \large \sin x = \pm \sqrt{\frac{5}{6}}]


http://www.wolframalpha.com/input/?i=6+sin%5E3+x+-+18+sin%5E2+x+%2B+5+sin+x+%2B+15+%3D+0 lists the real solutions to the original equation (including non-real roots).