Решение:

a)

$$x + 2 - \frac{x^2 + 4}{x - 2} = \frac{(x + 2)(x - 2) - (x^2 + 4)}{x - 2} = \frac{x^2 - 4 - x^2 - 4}{x - 2} = \frac{-8}{x - 2} = \frac{8}{2 - x}$$

Ответ: $$\frac{8}{2 - x}$$

б)

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

Ответ: $$\frac{(m + n)^2}{2mn(m - n)}$$

в)

$$\frac{p + 3}{p + 4} - \frac{p - 3}{p - 4} = \frac{(p + 3)(p - 4) - (p - 3)(p + 4)}{(p + 4)(p - 4)} = \frac{p^2 - 4p + 3p - 12 - (p^2 + 4p - 3p - 12)}{p^2 - 16} = \frac{p^2 - p - 12 - p^2 - p + 12}{p^2 - 16} = \frac{-2p}{p^2 - 16}$$

Ответ: $$\frac{-2p}{p^2 - 16}$$

г)

$$\frac{7p + q}{p^2 - pq} + \frac{p + 7q}{q^2 - pq} = \frac{7p + q}{p(p - q)} + \frac{p + 7q}{q(q - p)} = \frac{7p + q}{p(p - q)} - \frac{p + 7q}{q(p - q)} = \frac{q(7p + q) - p(p + 7q)}{pq(p - q)} = \frac{7pq + q^2 - p^2 - 7pq}{pq(p - q)} = \frac{q^2 - p^2}{pq(p - q)} = \frac{-(p^2 - q^2)}{pq(p - q)} = \frac{-(p - q)(p + q)}{pq(p - q)} = \frac{-(p + q)}{pq} = -\frac{p + q}{pq}$$

Ответ: $$\frac{-(p + q)}{pq}$$

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

Ответ: $$\frac{a - b}{a + b}$$