Let the complex number z be defined as:

z = $3 + 4i$ where $i$ is the imaginary unit, defined by $i^2 = -1$. Determine the value of the expression:

$$|z|^2 - 2 ext{Re}(z^2) + 5$$

where $|z|$ represents the modulus of the complex number z and $ ext{Re}(z^2)$ represents the real part of the square of z. Simplify your answer to the nearest hundredth.