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

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

$\frac{c + 7}{c - 7} + \frac{28c}{49 - c^2} = \frac{c + 7}{c - 7} - \frac{28c}{c^2 - 49} = \frac{c + 7}{c - 7} - \frac{28c}{(c - 7)(c + 7)} = \frac{(c + 7)(c + 7) - 28c}{(c - 7)(c + 7)} = \frac{(c + 7)^2 - 28c}{(c - 7)(c + 7)} = \frac{c^2 + 14c + 49 - 28c}{(c - 7)(c + 7)} = \frac{c^2 - 14c + 49}{(c - 7)(c + 7)} = \frac{(c - 7)^2}{(c - 7)(c + 7)} = \frac{c - 7}{c + 7}$

$\frac{5a + 3}{2a^2 + 6a} + \frac{6 - 3a}{a^2 - 9} = \frac{5a + 3}{2a(a + 3)} + \frac{6 - 3a}{(a - 3)(a + 3)} = \frac{(5a + 3)(a - 3) + 2a(6 - 3a)}{2a(a - 3)(a + 3)} = \frac{5a^2 + 3a - 15a - 9 + 12a - 6a^2}{2a(a - 3)(a + 3)} = \frac{-a^2 - 9}{2a(a^2 - 9)}$

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

$\frac{2p}{p - 5} - \frac{5}{p + 5} + \frac{2p^2}{25 - p^2} = \frac{2p}{p - 5} - \frac{5}{p + 5} - \frac{2p^2}{p^2 - 25} = \frac{2p}{p - 5} - \frac{5}{p + 5} - \frac{2p^2}{(p - 5)(p + 5)} = \frac{2p(p + 5) - 5(p - 5) - 2p^2}{(p - 5)(p + 5)} = \frac{2p^2 + 10p - 5p + 25 - 2p^2}{(p - 5)(p + 5)} = \frac{5p + 25}{(p - 5)(p + 5)} = \frac{5(p + 5)}{(p - 5)(p + 5)} = \frac{5}{p - 5}$

$\frac{1}{y} - \frac{y + 8}{16 - y^2} - \frac{2}{y - 4} = \frac{1}{y} + \frac{y + 8}{y^2 - 16} - \frac{2}{y - 4} = \frac{1}{y} + \frac{y + 8}{(y - 4)(y + 4)} - \frac{2}{y - 4} = \frac{y^2 - 16 + y(y + 8) - 2y(y + 4)}{y(y - 4)(y + 4)} = \frac{y^2 - 16 + y^2 + 8y - 2y^2 - 8y}{y(y - 4)(y + 4)} = \frac{-16}{y(y^2 - 16)} = \frac{16}{y(16 - y^2)}$

$\frac{2b - 1}{4b + 2} + \frac{4b}{4b^2 - 1} + \frac{2b + 1}{3 - 6b} = \frac{2b - 1}{4b + 2} + \frac{4b}{4b^2 - 1} - \frac{2b + 1}{6b - 3} = \frac{2b - 1}{2(2b + 1)} + \frac{4b}{(2b - 1)(2b + 1)} - \frac{2b + 1}{3(2b - 1)} = \frac{3(2b - 1)(2b - 1) + 6 * 4b - 2(2b + 1)(2b + 1)}{6(2b - 1)(2b + 1)} = \frac{3(2b - 1)^2 + 24b - 2(2b + 1)^2}{6(2b - 1)(2b + 1)} = \frac{3(4b^2 - 4b + 1) + 24b - 2(4b^2 + 4b + 1)}{6(2b - 1)(2b + 1)} = \frac{12b^2 - 12b + 3 + 24b - 8b^2 - 8b - 2}{6(2b - 1)(2b + 1)} = \frac{4b^2 + 4b + 1}{6(2b - 1)(2b + 1)} = \frac{(2b + 1)^2}{6(2b - 1)(2b + 1)} = \frac{2b + 1}{6(2b - 1)}$