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To solve the given problem, we need to evaluate the expression inside the absolute value and then take the magnitude of the resulting complex number.

The given expression is \(|2 + i^2 + 2i^3|\).

First, we need to simplify the powers of \(i\):

- \(i^2 = -1\)
- \(i^3 = i^2 \cdot i = -i\)

Substitute these values into the expression:

\[2 + i^2 + 2i^3 = 2 - 1 + 2(-i)\]
\[= 1 - 2i\]

Now, we need to find the magnitude of the complex number \(1 - 2i\). The magnitude \(|a + bi|\) of a complex number \(a + bi\) is given by:

\[|a + bi| = \sqrt{a^2 + b^2}\]

For \(1 - 2i\):

\[|1 - 2i| = \sqrt{1^2 + (-2)^2}\]
\[= \sqrt{1 + 4}\]
\[= \sqrt{5}\]

Thus, the answer is \(\sqrt{5}\).

The correct option is C. \(\sqrt{5}\).

Do you want more details or have any questions? Here are 5 questions you might want to ask next:
1. How do you simplify expressions involving imaginary numbers?
2. What are the basic properties of complex numbers?
3. How do you find the magnitude of a complex number?
4. Can you explain the different powers of \(i\)?
5. What is the geometric interpretation of complex numbers?

**Tip:** Always remember the basic identities for imaginary units: \(i^2 = -1\), \(i^3 = -i\), and \(i^4 = 1\). This helps in simplifying complex number expressions quickly.

Learn how to solve the problem |2 + i^2 + 2i^3| and find its magnitude step-by-step. Understand the concepts of complex numbers, absolute value, and the calculation of magnitudes. Suitable for advanced high school students.

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