<s>[INST] Solving $2\sin\theta\cos\theta + \sin\theta = 0$ 
The question is to solve the following question in the range $-\pi \le \theta \le  \pi$
$$2\sin\theta\cos\theta + \sin\theta = 0$$

I missed the obvious sin factorisation so proceeded as below. I see the correct solutions should be $\pm2/3\pi$ and the values when $\sin\theta = 0$. Although I missed the early factorisation I don't know what I'm doing to actually arrive at an incorrect answer:
$$\begin{align}
2\sin\theta\cos\theta + \sin\theta &= 0 \qquad\text{(square)} \tag{1} \\
4\sin^2\theta\cos^2\theta + \sin^2\theta &= 0 \tag{2}\\
4\sin^2\theta(1-\sin^2\theta) + \sin^2\theta &= 0 \tag{3} \\
4\sin^2\theta - 4\sin^4\theta + \sin^2\theta &= 0 \tag{4} \\
5\sin^2\theta  - 4\sin^4\theta &= 0 \tag{5}
\end{align}$$
and then solving by substitution/the quadratic equation I get $\sin\theta = \pm\sqrt(5)/2$ and $0$ but as this out of bounds for sin so cannot be the answer.
I know using Symbolab that this solution is correct for the quadratic I've generated, so I must be going wrong somewhere above after missing that factoristion. I feel like it is in the squaring step but not sure what would be wrong here...
Thanks a lot for your help.
 [/INST] I'll start by graphing this function x-axis is $\theta / \pi $ which shows the function is zero at 5 points.

\begin{align}
2 \cdot \sin(\theta)\cos(\theta) + \sin(\theta) & = 0 \\
sin(\theta) \cdot (2\cdot\cos(\theta)+1) & = 0 
\end{align}
So either $\sin(\theta) = 0$ or $2\cdot\cos(\theta)+1 = 0 \Rightarrow \cos(\theta) = 
-0.5$
Considering each of these cases then $\theta$ is $-\pi$, $-\dfrac{2}{3}\pi$, $0$, $\dfrac{2}{3}\pi$ or $\pi$
</s>