To simplify the given expression:
$\frac{p - q}{p + q} + \frac{p + q}{p - q}$

Step 1: Find the common denominator

The least common denominator (LCD) for $\frac{p - q}{p + q}$ and $\frac{p + q}{p - q}$ is $(p + q)(p - q)$.

Rewriting each fraction with this common denominator:
$\frac{(p - q)^2}{(p + q)(p - q)} + \frac{(p + q)^2}{(p - q)(p + q)} = \frac{(p - q)^2 + (p + q)^2}{(p + q)(p - q)}$

Step 2: Expand the numerators

Use the square of a binomial identity $(a \pm b)^2 = a^2 \pm 2ab + b^2$.

1. $(p - q)^2 = p^2 - 2pq + q^2$
2. $(p + q)^2 = p^2 + 2pq + q^2$

Now, add the two results:
$(p - q)^2 + (p + q)^2 = (p^2 - 2pq + q^2) + (p^2 + 2pq + q^2)$

Combine like terms:
$p^2 + p^2 - 2pq + 2pq + q^2 + q^2 = 2p^2 + 2q^2$

Step 3: Simplify the entire fraction

The fraction becomes:
$\frac{2p^2 + 2q^2}{(p + q)(p - q)} = \frac{2(p^2 + q^2)}{(p + q)(p - q)}$

Final Answer

The simplified form of the given expression is:
$\frac{2(p^2 + q^2)}{(p + q)(p - q)}$

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Would you like a more detailed explanation for any of the steps?
Here are 5 related questions to expand your understanding:

1. How do you find the least common denominator of two rational expressions?
2. What are the identities for $(a + b)^2$ and $(a - b)^2$?
3. How do you factor quadratic expressions like $p^2 - q^2$?
4. Can the final result $\frac{2(p^2 + q^2)}{(p + q)(p - q)}$ be simplified further?
5. What is the geometric interpretation of the difference of squares?

Tip: When dealing with rational expressions, always look for ways to factor the numerator and denominator. This approach often reveals common factors that can be canceled out.