To determine which expression represents the sum of two odd integers, let’s first understand the properties of odd integers:

1. An odd integer can be expressed as $2n + 1$, where $n$ is an integer.
2. The sum of two odd integers, say $(2a + 1) + (2b + 1)$, results in: $(2a + 1) + (2b + 1) = 2a + 2b + 2 = 2(a + b + 1)$ This final result is an even integer.

Now, let's examine each option to see which one aligns with the properties of odd integers:

1. Option A: $2(a + b) + 1$

   • This expression is of the form $2k + 1$, where $k = a + b$, representing an odd integer.

2. Option B: $a + b$

   • If $a$ and $b$ are both odd integers, their sum would be even (as shown above).

3. Option C: $a + b + 2$

   • This expression is an even integer if $a + b$ is even (sum of two odd integers is even, so $a + b + 2$ is still even).

4. Option D: $2(a + b + 1)$

   • This is clearly an even integer, as it’s a multiple of 2.

Based on this analysis, the correct answer is:

A. $2(a + b) + 1$

This expression represents an odd integer, and it is the sum of two odd integers.

Would you like further explanation or have any questions?

Here are five related questions to explore further:

1. How would you represent the sum of two even integers using variables?
2. Why does the sum of two odd integers result in an even integer?
3. Can you derive an expression for the product of two odd integers?
4. How would you represent the difference between two odd integers?
5. What would be the sum of an odd and an even integer?

Tip: Remember, the sum of two odd integers is always even, while the sum of an odd and even integer is always odd.