$\frac{p^2 - q^2}{p^2} * \frac{pq + q^2}{(p + q)^2} = \frac{(p - q)(p + q)}{p^2} * \frac{q(p + q)}{(p + q)^2} = \frac{(p - q)(p + q)}{p^2} * \frac{q}{p + q} = \frac{p - q}{p^2} * \frac{q}{1} = \frac{q(p - q)}{p^2}$

$\frac{a^2 - 9b^2}{a^2 - ab} : \frac{a^2 + 3ab}{a - b} = \frac{a^2 - 9b^2}{a^2 - ab} * \frac{a - b}{a^2 + 3ab} = \frac{(a - 3b)(a + 3b)}{a(a - b)} * \frac{a - b}{a(a + 3b)} = \frac{a - 3b}{a} * \frac{1}{a} = \frac{a - 3b}{a^2}$

$\frac{3x^2 - 3y^2}{x^2 + xy} * \frac{x + y}{6x - 6y} = \frac{3(x^2 - y^2)}{x(x + y)} * \frac{x + y}{6(x - y)} = \frac{3(x - y)(x + y)}{x(x + y)} * \frac{x + y}{6(x - y)} = \frac{3(x - y)}{x} * \frac{x + y}{6(x - y)} = \frac{1}{x} * \frac{x + y}{2} = \frac{x + y}{2x}$

$\frac{m^2 - n^2}{(m + n)^2} : \frac{4m - 4n}{3m + 3n} = \frac{m^2 - n^2}{(m + n)^2} * \frac{3m + 3n}{4m - 4n} = \frac{(m - n)(m + n)}{(m + n)^2} * \frac{3(m + n)}{4(m - n)} = \frac{m - n}{m + n} * \frac{3(m + n)}{4(m - n)} = \frac{3}{4}$