The number of elements in the set S = θ ∈ [- 4π, 4π]: 3 ^ 2 2θ + 6...

$3\cos^{2}2\theta + 6\cos 2\theta - 10\cos^{2}\theta + 5 = 0$

$3\cos^{2}2\theta + 6\cos 2\theta - 5\left(1 + \cos 2\theta\right) + 5 = 0$

$\cos 2\theta\left(3\cos 2\theta + 1\right) = 0$

$\cos 2\theta = 0$ or $\cos 2\theta = - \dfrac{1}{3}$

Given $\theta \in \left[- 4\pi, 4\pi\right] ⇒ 2\theta \in \left[- 8\pi, 8\pi\right]$

So for $2\theta = \left(2n + 1\right)\dfrac{\pi}{2}$

$\theta = \pm \dfrac{\pi}{4}, \pm \dfrac{3\pi}{4}, \ldots \ldots \ldots \pm 15\dfrac{\pi}{4}$ gives $16$ solution

Similarly $\cos 2\theta = - \dfrac{1}{3}$ gives $16$ solution

hence, total elements in set $S$ will be $32$