Value of Absolute Square of Complex Number with Real Fraction

Let the given expression be denoted as $w = \frac{2 + 3z + 4z^2}{2 - 3z + 4z^2} = 1 + \frac{6}{\frac{2}{z} - 3 + 4z}$. For $w$ to be a real number, the quantity $\frac{2}{z} + 4z$ must also be real, which implies $\frac{2}{z} + 4z = \frac{2}{\bar{z}} + 4\bar{z}$. Rearranging this equality gives $(z - \bar{z})\left(\frac{2}{|z|^2} - 4\right) = 0$. Since $z$ has a non-zero imaginary part, $z - \bar{z} \neq 0$, leading directly to $|z|^2 = \frac{1}{2} = 0.5$.