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$\frac{a^2+b^2}{2a^2+2ab}+\frac{b}{a+b}=\frac{a^2+b^2}{2a(a+b)}+\frac{b}{a+b}=\frac{a^2+b^2}{2a(a+b)}+\frac{2a*b}{2a(a+b)}=\frac{a^2+b^2+2ab}{2a(a+b)}=\\\ \\\ =\frac{(a+b)^2}{2a(a+b)}=\frac{a+b}{2a}$

$\frac{4y}{y^2-x^2}-\frac{2}{y-x}=\frac{4y}{(y-x)(y+x)}-\frac{2}{y-x}=\\\ \\\ =\frac{4y}{(y-x)(y+x)}-\frac{2(y+x)}{(y-x)(y+x)}=\frac{4y-2(y+x)}{(y-x)(y+x)}=\frac{4y-2y-2x}{(y-x)(y+x)}=\\\ \\\ =\frac{2y-2x}{(y-x)(y+x)}=\frac{2(y-x)}{(y-x)(y+x)}=\frac{2}{y+x}$