I know the modus operandi,
given $\sin x + \sin 2x = 0$, (range of x = $[0, \pi]$)
1st, rewrite the equation: $\sin x = -\sin 2x$.
2nd, on the cartesian plane, plot the first sine out, and do the same thing for the other one.
3rd, find the intercepts between the first and the second function.
I've followed this algorithm, and I've figured out that this equation has at most two solutions (0 and $\pi$), but the multiple choice question said otherwise.
(1) it has exactly one solution.
(2) it has exactly 3 solutions.
(3) it has infinite solutions.
(4) it has exactly 2 solutions.
the correct answer (according to exercise) is (2).
EDIT (correct solution):
the modus operandi is correct, the problem is that I didn't consider ($\pi/4 \space and \space 3/2\space\pi)$. Meaning, half of $\pi/2$ and $\pi/2 + \space half$. Personally, I prefer using this method over the others.