Answer: 540

Solution:

Write $z = 4(\cos\theta + i\sin\theta),$ so $\frac{1}{z} = \frac{1}{4}(\cos\theta - i\sin\theta).$ The real part of $(75 + 117i)z$ is $4(75\cos\theta - 117\sin\theta)$ $= 300\cos\theta - 468\sin\theta,$ and the real part of $(96 + 144i) \cdot \frac{1}{4}(\cos\theta - i\sin\theta)$ is $24\cos\theta + 36\sin\theta.$

The total real part is $324\cos\theta - 432\sin\theta,$ whose maximum over $\theta$ is $$\begin{aligned} &\sqrt{324^2 + 432^2} \\ &= 108\sqrt{3^2 + 4^2} \\ &= 108 \cdot 5 = 540. \end{aligned}$$