$\frac{x^2 + 14x + 49}{x + 6} : (\frac{13}{x + 6} - x + 6) = \frac{(x + 7)^2}{x + 6} : \frac{13 - x(x + 6) + 6(x + 6)}{x + 6} = \frac{(x + 7)^2}{x + 6} : \frac{13 - x^2 - 6x + 6x + 36}{x + 6} = \frac{(x + 7)^2}{x + 6} : \frac{-x^2 + 49}{x + 6} = \frac{(x + 7)^2}{x + 6} * \frac{x + 6}{49 - x^2} = \frac{(x + 7)^2}{1} * \frac{1}{(7 - x)(7 + x)} = \frac{7 + x}{7 - x}$

$(c - \frac{2c - 9}{c + 8}) : \frac{c^2 + 3c}{c^2 - 64} + \frac{24}{c} = \frac{c(c + 8) - (2c - 9)}{c + 8} : \frac{c(c + 3)}{(c - 8)(c + 8)} + \frac{24}{c} = \frac{c^2 + 8c - 2c + 9}{c + 8} * \frac{(c - 8)(c + 8)}{c(c + 3)} + \frac{24}{c} = \frac{c^2 + 6c + 9}{1} * \frac{c - 8}{c(c + 3)} + \frac{24}{c} = \frac{(c + 3)^2}{1} * \frac{c - 8}{c(c + 3)} + \frac{24}{c} = \frac{c + 3}{1} * \frac{c - 8}{c} + \frac{24}{c} = \frac{(c + 3)(c - 8)}{c} + \frac{24}{c} = \frac{c^2 + 3c - 8c - 24 + 24}{c} = \frac{c^2 - 5c}{c} = \frac{c(c - 5)}{c} = c - 5$

$(\frac{36}{x^2 - 9} - \frac{x - 3}{x + 3} - \frac{3 + x}{3 - x}) : \frac{6}{3 - x} = (\frac{36}{(x - 3)(x + 3)} - \frac{x - 3}{x + 3} + \frac{3 + x}{x - 3}) * \frac{3 - x}{6} = \frac{36 - (x - 3)^2 + (x + 3)^2}{(x - 3)(x + 3)} * (-\frac{x - 3}{6}) = \frac{36 - (x^2 - 6x + 9) + x^2 + 6x + 9}{x + 3} * (-\frac{1}{6}) = \frac{36 - x^2 + 6x - 9 + x^2 + 6x + 9}{x + 3} * (-\frac{1}{6}) = \frac{12x + 36}{x + 3} * (-\frac{1}{6}) = \frac{12(x + 3)}{x + 3} * (-\frac{1}{6}) = -2$

$(\frac{2y - 1}{y^2 + 2y + 4} + \frac{9y + 6}{y^3 - 8} + \frac{1}{y - 2}) * \frac{y^2 - 4}{18} = (\frac{2y - 1}{y^2 + 2y + 4} + \frac{9y + 6}{(y - 2)(y^2 + 2y + 4)} + \frac{1}{y - 2}) * \frac{(y - 2)(y + 2)}{18} = \frac{(2y - 1)(y - 2) + 9y + 6 + y^2 + 2y + 4}{(y - 2)(y^2 + 2y + 4)} * \frac{(y - 2)(y + 2)}{18} = \frac{2y^2 - y - 4y + 2 + 11y + 10 + y^2}{y^2 + 2y + 4} * \frac{y + 2}{18} = \frac{3y^2 + 6y + 12}{y^2 + 2y + 4} * \frac{y + 2}{18} = \frac{3(y^2 + 2y + 4)}{y^2 + 2y + 4} * \frac{y + 2}{18} = \frac{y + 2}{6}$