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

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

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

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