Answer
$$\theta =10^{\circ}+120^{\circ}k, \ \ \text{or} \ \ \theta = 50^{\circ}+120^{\circ}k, \ \ \text{or} \ \ 90^{\circ}+120^{\circ}k $$
Work Step by Step
Given $$2\sin^23\theta+\sin 3\theta -1=0$$
Since
\begin{align*}
2\sin^23\theta+\sin 3\theta -1&=0\\
(2\sin 3\theta-1)(\sin3\theta+1)&=0
\end{align*}
Then $$2\sin 3\theta-1=0\ \Rightarrow \ \sin3\theta =\frac{1}{2}\ \Rightarrow \ \ 3\theta =30^{\circ}+360^{\circ}k,\ 150^{\circ}+360^{\circ}k$$
Hence $$\theta =10^{\circ}+120^{\circ}k, \ \ \text{or} \ \ \theta = 50^{\circ}+120^{\circ}k$$
or $$\sin3\theta+1=0 \ \Rightarrow \ \ \sin3\theta= -1\ \Rightarrow \ \ 3\theta =270^{\circ}+360^{\circ}k$$
Hence
$$ \theta =90^{\circ}+120^{\circ}k$$
where $k$ is an integer