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The problem is asking which expression demonstrates the **commutative property of addition** in the first step of simplifying the expression \((-1 + i) + (2i + 5i)\).

### **Commutative Property of Addition**
The commutative property states that the order in which two numbers are added does not affect the sum. That is:
\[
a + b = b + a
\]

### Given Expression
The original expression is \((-1 + i) + (2i + 5i)\). According to the commutative property, the terms can be rearranged.

Let's look at the provided options:

1. **First Option:** \((-1 + i) + (2i + 5i)\)
   - This is the original expression and does not yet demonstrate the commutative property since nothing has been rearranged.

2. **Second Option:** \(-1 + (i + 2i) + 5i\)
   - In this option, \(i + 2i\) has been grouped together, but there is no clear commutative swap.

3. **Third Option:** \((-1 + 2i) + (i + 5i)\)
   - This shows that terms have been rearranged. The \(2i\) and \(i\) have swapped places, demonstrating the commutative property.

4. **Fourth Option:** \(-(-1 - i) + (2i + 5)\)
   - This does not represent a simple commutative swap.

### **Conclusion**
The **third option** \((-1 + 2i) + (i + 5i)\) correctly demonstrates the **commutative property** of addition, as the order of terms has been swapped.

Do you have any questions or would you like further details?

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Here are five related questions to expand your understanding:

1. How does the commutative property apply to multiplication?
2. What is the associative property, and how does it differ from the commutative property?
3. Can the commutative property be applied to subtraction or division?
4. How does the commutative property work in complex numbers?
5. What is the next step after applying the commutative property to simplify the expression?

**Tip:** Always look for opportunities to simplify terms by combining like terms before using properties such as commutative or associative.

Which demonstrates commutative property of addition in the first step of simplifying the expression (-1 + i) + (2i + 5i)?

Understand how the commutative property of addition is applied in simplifying complex number expressions such as (-1 + i) + (2i + 5i). Explore key algebraic concepts related to complex numbers and the commutative property. Suitable for high school students in Grades 9-12.

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