Let's evaluate the expression step-by-step. The expression is:

\[
I + (2 \cdot I^2) + (3 \cdot I^3) + 4
\]

where \( I \) is the imaginary unit, defined as \( I = \sqrt{-1} \).

1. **Calculate \( I \):**
   \[
   I = \sqrt{-1}
   \]

2. **Calculate \( I^2 \):**
   \[
   I^2 = -1
   \]

3. **Calculate \( I^3 \):**
   \[
   I^3 = I^2 \cdot I = (-1) \cdot I = -I
   \]

Now, substitute these values into the expression:

\[
I + (2 \cdot I^2) + (3 \cdot I^3) + 4
= I + (2 \cdot -1) + (3 \cdot -I) + 4
\]

Simplify each term:

- \( I \)
- \( 2 \cdot -1 = -2 \)
- \( 3 \cdot -I = -3I \)
- \( 4 \)

Combine all terms:

\[
I - 2 - 3I + 4
\]

Group the real and imaginary parts:

\[
(4 - 2) + (I - 3I) = 2 - 2I
\]

So, the simplified answer is:

\[
2 - 2I
\]

Would you like further details or have any questions?

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Here are some related questions for further exploration:
1. What are the higher powers of \( I \) (e.g., \( I^4, I^5 \)) and how do they behave?
2. How does the imaginary unit \( I \) influence complex number calculations?
3. What is the modulus (absolute value) of a complex number like \( 2 - 2I \)?
4. How can complex numbers be represented on the complex plane?
5. What are practical applications of complex numbers in engineering or physics?

**Tip:** Remember, powers of \( I \) repeat every four terms: \( I, -1, -I, 1 \).

Explore the step-by-step solution for simplifying the complex expression I + 2I^2 + 3I^3 + 4. This problem covers key concepts in complex numbers, including the properties of the imaginary unit I and algebraic simplification techniques. Suitable for students in Grades 10-12.

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