Simplify the following:

(A) $\frac{4x^2 + 4x + 1}{4x^2 - 1}$

(B) $\frac{9(3a^3 - 24b^3)}{9a^2 - 36b^2}$

(C) $\frac{s^3 + 125t^3}{s^2 - 2st - 35t^2}$

Note: Assume that the denominators are not equal to 0.

$$(A)\ \frac{4x^2 + 4x + 1}{4x^2 - 1}$$

$$\text{Numerator} = (2x + 1)^2$$

$$\text{Denominator} = (2x - 1)(2x + 1)$$

Required expression in simplified form is:

$$\frac{(2x + 1)}{(2x - 1)}$$

$$(B)\ \frac{9(3a^3 - 24b^3)}{9a^2 - 36b^2}$$

$$\text{Numerator} = 9 \times 3(a^3 - 8b^3)$$

$$= 27(a - 2b)(a^2 + 2ab + 4b^2)$$

$$\text{Denominator} = 9(a^2 - 4b^2)$$

$$= 9(a - 2b)(a + 2b)$$

Required expression in simplified form is:

$$\frac{27(a^2 + 2ab + 4b^2)}{[9(a + 2b)]} = \frac{3(a^2 + 2ab + 4b^2)}{(a + 2b)}$$

$$(C)\ \text{Numerator} = s^3 + (5t)^3$$

$$= (s + 5t)(s^2 - 5st + 25t^2)$$

$$\text{Denominator} = s^2 - 2st - 35t^2$$

$$\text{Split: Use } a + b = -2t,\ ab = -35t^2$$

$$\text{So, } a = -7t,\ b = 5t.$$

$$\text{Check: } -7t + 5t = -2t,\ (-7t)(5t) = -35t^2$$

$$s^2 + 5st - 7st - 35t^2 = s(s + 5t) - 7t(s + 5t)$$

$$= (s + 5t)(s - 7t)$$

Required expression in simplified form is:

$$\frac{(s^2 - 5st + 25t^2)}{(s - 7t)}$$