58. Упростите выражение: 1) $$\frac{a^2 + b^2}{2a^2 + 2ab} + \frac{b}{a + b}$$; 2) $$\frac{a^2 + 9}{a^2 - 9} - \frac{a}{a + 3}$$; 3) $$\frac{c + 1}{3c} : \frac{c^2 - 1}{6c^2}$$; 4) $$\frac{m^2 + 2mn + n^2}{m^2 - n^2} : (m + n)$$.

1) $$\frac{a^2 + b^2}{2a^2 + 2ab} + \frac{b}{a + b} = \frac{a^2 + b^2}{2a(a + b)} + \frac{b}{a + b} = \frac{a^2 + b^2 + 2ab}{2a(a + b)} = \frac{(a + b)^2}{2a(a + b)} = \frac{a + b}{2a}$$ 2) $$\frac{a^2 + 9}{a^2 - 9} - \frac{a}{a + 3} = \frac{a^2 + 9}{(a - 3)(a + 3)} - \frac{a}{a + 3} = \frac{a^2 + 9 - a(a - 3)}{(a - 3)(a + 3)} = \frac{a^2 + 9 - a^2 + 3a}{(a - 3)(a + 3)} = \frac{3a + 9}{(a - 3)(a + 3)} = \frac{3(a + 3)}{(a - 3)(a + 3)} = \frac{3}{a - 3}$$ 3) $$\frac{c + 1}{3c} : \frac{c^2 - 1}{6c^2} = \frac{c + 1}{3c} * \frac{6c^2}{c^2 - 1} = \frac{c + 1}{3c} * \frac{6c^2}{(c - 1)(c + 1)} = \frac{6c^2(c + 1)}{3c(c - 1)(c + 1)} = \frac{2c}{c - 1}$$ 4) $$\frac{m^2 + 2mn + n^2}{m^2 - n^2} : (m + n) = \frac{(m + n)^2}{(m - n)(m + n)} : (m + n) = \frac{(m + n)^2}{(m - n)(m + n)} * \frac{1}{m + n} = \frac{(m + n)^2}{(m - n)(m + n)^2} = \frac{1}{m - n}$$