$\frac{a^2-b^2}{2a^{2}b}\ast \frac{4a{b}^2}{a+b}=\frac{(a-b)(a+b)}{2a^{2}b}\ast \frac{4a{b}^2}{a+b}=\frac{a-b}{a}\ast \frac{2b}{1}=\frac{2b(a-b)}{a}$

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

$\frac{mn-m^2}{2m}\ast \frac{8n}{n^2-m^2}=\frac{m(n-m)}{2m}\ast \frac{8n}{(n-m)(n+m)}=\frac{n-m}{2}\ast \frac{8n}{(n-m)(n+m)}=\frac{1}{1}\ast \frac{4n}{n+m}=\frac{4n}{n+m}$

$\frac{2a-4}{b+1}:\frac{a^2-4}{(b+1{)}^2}=\frac{2a-4}{b+1}\ast \frac{(b+1{)}^2}{a^2-4}=\frac{2(a-2)}{b+1}\ast \frac{(b+1{)}^2}{(a-2)(a+2)}=\frac{2}{1}\ast \frac{b+1}{a+2}=\frac{2(b+1)}{a+2}$

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

$\frac{16-m^2}{m^2-3m}:\frac{m^2+4m}{m^2-9}=\frac{16-m^2}{m^2-3m}\ast \frac{m^2-9}{m^2+4m}=\frac{(4-m)(4+m)}{m(m-3)}\ast \frac{(m-3)(m+3)}{m(m+4)}=\frac{4-m}{m}\ast \frac{m+3}{m}=\frac{(4-m)(m+3)}{m^2}$