$\frac{p - t + 2pt - 2t^2}{1 + 4t + 4t^2} = \frac{(p - t) + (2pt - 2t^2)}{(1 + 2t)^2} = \frac{(p - t) + 2t(p - t)}{(1 + 2t)^2} = \frac{(p - t)(1 + 2t)}{(1 + 2t)^2} = \frac{p - t}{1 + 2t}$

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

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

$\frac{6k + 5l + 6k^2 + 5kl}{k^3 + 1} = \frac{(6k + 6k^2) + (5l + 5kl)}{(k + 1)(k^2 - k + 1)} = \frac{6k(1 + k) + 5l(1 + k)}{(k + 1)(k^2 - k + 1)} = \frac{(1 + k)(6k + 5l)}{(k + 1)(k^2 - k + 1)} = \frac{6k + 5l}{k^2 - k + 1}$