To solve this trigonometric equation, we can use the fact that sine squared plus cosine squared equals 1. However, the given equation is not in that form. We can rewrite the equation in terms of sine and cosine squared as follows:

3sin^2 x + 11sin x cos x + 10cos^2 x = 0

Using the trigonometric identity sin^2 x + cos^2 x = 1, we can substitute cos^2 x with 1 - sin^2 x:

3sin^2 x + 11sin x (1 - sin^2 x) + 10(1 - sin^2 x) = 0

Expanding and simplifying, we get:

3sin^2 x + 11sin x - 11sin^3 x + 10 - 10sin^2 x = 0
-11sin^3 x - 7sin^2 x + 11sin x + 10 = 0

Now, we have a cubic equation in terms of sin x. This equation can be solved either numerically or graphically to find the values of sin x that satisfy the equation.