Modulus of a Complex Number Expression

Consider the complex number $z = 3 - 4i$. You are tasked with finding the modulus of $z^2 + 2z + 5$. First, calculate $z^2$:

$$z^2 = (3 - 4i)^2 = 9 - 24i - 16(-1) = 9 - 24i + 16 = 25 - 24i$$

Next, add $2z + 5$ to $z^2$:

$$2z = 2(3 - 4i) = 6 - 8i$$

So, we now compute:

$$z^2 + 2z + 5 = (25 - 24i) + (6 - 8i) + 5 = 36 - 32i$$

Finally, determine the modulus of $36 - 32i$. The modulus is given by the formula:

$$|a + bi| = ext{sqrt}(a^2 + b^2)$$ where $a = 36$ and $b = -32$. Calculate:

$$|36 - 32i| = ext{sqrt}(36^2 + (-32)^2)$$