Answer
$$\theta =70^{\circ}+120^{\circ}k, \ \ \text{or} \ \ \theta = 110^{\circ}+120^{\circ}k,\ \ \text{or} \ \ \theta=90^{\circ}+120^{\circ}k $$
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 $$2\sin 3\theta+1=0\ \ \Rightarrow \ \sin3\theta =\frac{-1}{2} \\ \Rightarrow \ \ \ \ 3\theta =210^{\circ}+360^{\circ}k,\ \ \ 330^{\circ}+360^{\circ}k$$
Hence $$\theta =70^{\circ}+120^{\circ}k, \ \ \text{or} \ \ \theta = 110^{\circ}+120^{\circ}k$$
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