To simplify rational expressions, we factorise the numerator and denominator and cancel the common factors.

For (i), the numerator $4x^{2}$ + 4x + 1 is a perfect square, (2x + 1)^2, and the denominator $4x^{2}$ - 1 is a difference of squares, (2x + 1)(2x - 1). Cancelling (2x + 1) gives (2x + 1) / (2x - 1).

For (ii), the numerator simplifies to 27(a - 2b)($a^{2}$ + 2ab + $4b^{2}$) and the denominator to 9(a - 2b)(a + 2b). Cancelling 9(a - 2b) yields 3($a^{2}$ + 2ab + $4b^{2}$) / (a + 2b).

For (iii), the numerator is a sum of cubes, (s + 5t)($s^{2}$ - 5st + $25t^{2}$), and the denominator factors to (s + 5t)(s - 7t). Cancelling (s + 5t) leaves ($s^{2}$ - 5st + $25t^{2}$) / (s - 7t).