$\frac{3qp^2 - 27q}{27q - p^3q} = \frac{3q(p^2 - 9)}{q(27 - p^3)} = \frac{3q(p - 3)(p + 3)}{q(3 - p)(9 + 3p + p^2)} = -\frac{3(p - 3)(p + 3)}{(p - 3)(9 + 3p + p^2)} = -\frac{3(p + 3)}{9 + 3p + p^2}$

$\frac{x^6 - y^6}{x^3 + y^3} = \frac{(x^3 - y^3)(x^3 + y^3)}{x^3 + y^3} = x^3 - y^3$

$\frac{8mn^2 - 2m}{8mn^4 + mn} = \frac{2m(4n^2 - 1)}{mn(8n^3 + 1)} = \frac{2(2n - 1)(2n + 1)}{n(2n + 1)(4n^2 - 2n + 1)} = \frac{2(2n - 1)}{n(4n^2 - 2n + 1)}$

$\frac{y^6 + y^3}{y^6 - 1} = \frac{y^3(y^3 + 1)}{(y^3 - 1)(y^3 + 1)} = \frac{y^3}{y^3 - 1}$