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To solve the problem, we need to find the arithmetic mean of the expressions \((m + n)^2\) and \((m - n)^2\).

### Step 1: Expand both expressions
1. **Expand** \((m + n)^2\):
   \[
   (m + n)^2 = m^2 + 2mn + n^2
   \]
2. **Expand** \((m - n)^2\):
   \[
   (m - n)^2 = m^2 - 2mn + n^2
   \]

### Step 2: Find the arithmetic mean
The arithmetic mean of two numbers \(a\) and \(b\) is given by:
\[
\text{Arithmetic Mean} = \frac{a + b}{2}
\]
Substituting the expressions from Step 1:
\[
\text{Arithmetic Mean} = \frac{(m^2 + 2mn + n^2) + (m^2 - 2mn + n^2)}{2}
\]

### Step 3: Simplify the expression
Add the expanded forms:
\[
\text{Arithmetic Mean} = \frac{m^2 + 2mn + n^2 + m^2 - 2mn + n^2}{2}
\]
Simplify by combining like terms:
\[
\text{Arithmetic Mean} = \frac{2m^2 + 2n^2}{2} = m^2 + n^2
\]

### Conclusion:
The correct answer is \(\boxed{d}\), which corresponds to \(m^2 + n^2\).

Would you like any more details or explanations?

Here are some related questions you might find interesting:
1. What is the geometric mean of \((m + n)^2\) and \((m - n)^2\)?
2. How would the arithmetic mean change if \(m\) and \(n\) were negative?
3. What is the relationship between the arithmetic mean and the quadratic mean of two numbers?
4. How can you generalize this problem to find the arithmetic mean of \((m + kn)^2\) and \((m - kn)^2\)?
5. How does this result connect to the binomial theorem?

**Tip:** When dealing with algebraic expressions, expanding and simplifying step by step is key to avoiding errors and ensuring a clear solution.

Find the arithmetic mean of (m + n)^2 and (m - n)^2.

This problem demonstrates how to find the arithmetic mean of the expressions (m + n)^2 and (m - n)^2. The solution includes expanding both expressions and simplifying the result. Suitable for students in Grades 9-11 studying algebra.

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