To simplify the given expression:

(P^2 - q^2) / (P^2) * pq + q^2 / (P+q)^2

First, factor the numerator of the first fraction:

[(P+q)(P-q)] / (P^2) * pq + q^2 / (P+q)^2

Now, multiply the fractions:

[[(P+q)(P-q) * pq / (P^2)] + (q^2 / (P+q)^2]

Since the denominators are not the same, we need to find a common denominator. To do so, we will multiply the numerator and denominator of the first fraction by (P+q) to get a common denominator:

[((P+q)(P-q) * pq) / (P^2(P+q))] + (q^2 / (P+q)^2)

Expanding the first fraction:

[(P^2 - q^2) * pq / (P^2)(P+q)] + (q^2 / (P+q)^2)

Now, cancel out the common factors:

[(P^2q - q^3) / (P^3 + P^2q)] + (q^2 / (P+q)^2)

Therefore, the simplified expression is:

(P^2q - q^3) / (P^3 + P^2q) + (q^2 / (P+q)^2)