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

2)
$(5+ \frac{a}{b} ) * \frac{b}{(5b+a)^2} = \frac{5b +a}{b} * \frac{b}{(5b+a)^2} = \frac{b(5a+b)}{b(5a+b)(5a+b)} = \frac{1}{5a+b}$

3)
$\frac{a-3}{a+3} *(a+ \frac{a^2}{3-a}) = \frac{a-3}{a+3} * (\frac{a(a-3)}{a-3} - \frac{a^2}{a-3} ) = \frac{a-3}{a+3} * \frac{a^2-3a-a^2}{a-3} = \\ \\ = \frac{(a-3)*(-3a)}{(a+3)(a-3)} = \frac{1*(-3a)}{a+3} = - \frac{3a}{a+3}$

4) Как-то плохо сократилась дробь:
$\frac{x}{x-y} - \frac{x}{y+x} * \frac{(x+y)^2}{2x} = \frac{x}{x-y} - \frac{x(x+y)(x+y)}{2x(x+y)}= \frac{x}{x-y} - \frac{x+y}{2} = \\ \\ = \frac{x*2 -(x+y)(x-y)}{2(x-y)} = \frac{2x-(x^2-y^2)}{2(x-y)} = \frac{2x-x^2+y^2}{2(x-y)} = \frac{-x^2+2x +y^2}{2x-2y}$