$\frac{7a^2}{a^2 - 25} * \frac{5 - a}{a} = \frac{7a}{(a - 5)(a + 5)} * \frac{5 - a}{1} = -\frac{7a}{(5 - a)(a + 5)} * \frac{5 - a}{1} = -\frac{7a}{a + 5} * \frac{1}{1} = -\frac{7a}{a + 5}$

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

$\frac{a^4 - 1}{a^3 - a} * \frac{a}{1 + a^2} = \frac{(a^2 - 1)(a^2 + 1)}{a(a^2 - 1)} * \frac{a}{1 + a^2} = \frac{1}{1} * \frac{1}{1} = 1$

$\frac{a^2 - 8ab}{12b} : \frac{8b^2 - ab}{24a} = \frac{a(a - 8b)}{12b} : \frac{b(8b - a)}{24a} = \frac{a(a - 8b)}{12b} * \frac{24a}{b(8b - a)} = \frac{a(a - 8b)}{12b} * (-\frac{24a}{b(a - 8b)}) = \frac{a}{b} * (-\frac{2a}{b}) = -\frac{2a^2}{b^2}$

$\frac{5m^2 - 5n^2}{m^2 + n^2} : \frac{15n - 15m}{4m^2 + 4n^2} = \frac{5(m^2 - n^2)}{m^2 + n^2} : \frac{15(n - m)}{4(m^2 + n^2)} = \frac{5(m - n)(m + n)}{m^2 + n^2} : (-\frac{15(m - n)}{4(m^2 + n^2)}) = \frac{5(m - n)(m + n)}{m^2 + n^2} * (-\frac{4(m^2 + n^2)}{15(m - n)}) = \frac{m + n}{1} * (-\frac{4}{3}) = -\frac{4(m + n)}{3}$

$\frac{mn^2 - 36m}{m^3 - 8} : \frac{2n + 12}{6m - 12} = \frac{m(n^2 - 36)}{(m - 2)(m^2 + 2m + 4)} : \frac{2(n + 6)}{6(m - 2)} = \frac{m(n - 6)(n + 6)}{(m - 2)(m^2 + 2m + 4)} * \frac{6(m - 2)}{2(n + 6)} = \frac{m(n - 6)}{m^2 + 2m + 4} * \frac{3}{1} = \frac{3m(n - 6)}{m^2 + 2m + 4}$

$\frac{a^4 - 1}{a^2 - a + 1} : \frac{a - 1}{a^3 + 1} = \frac{(a^2 - 1)(a^2 + 1)}{a^2 - a + 1} : \frac{a - 1}{(a + 1)(a^2 - a + 1)} = \frac{(a - 1)(a + 1)(a^2 + 1)}{a^2 - a + 1} * \frac{(a + 1)(a^2 - a + 1)}{a - 1} = \frac{(a + 1)(a^2 + 1)}{1} * \frac{a + 1}{1} = (a + 1)^2(a^2 + 1)$

$\frac{4x^2 - 100}{6x} : (2x^2 - 20x + 50) = \frac{4(x^2 - 25)}{6x} : 2(x^2 - 10x + 25) = \frac{2(x - 5)(x + 5)}{3x} : 2(x - 5)^2 = \frac{2(x - 5)(x + 5)}{3x} * \frac{1}{2(x - 5)^2} = \frac{x + 5}{3x} * \frac{1}{x - 5} = \frac{x + 5}{3x(x - 5)}$