Consider the following statements:
(i) When $(0 < x < 1)$, then $(\frac{1}{1+x} < 1 - x + x^2)$
(ii) When $(0 < x < 1)$, then $(\frac{1}{1+x} > 1 - x + x^2)$
(iii) When $(-1 < x < 0)$, then $(\frac{1}{1+x} < 1 - x + x^2)$
(iv) When $(-1 < x < 0)$, then $(\frac{1}{1+x} > 1 - x + x^2)$
Then the correct statements are:

The easiest way to solve this question is by value-putting. Let's plug in some values in each of the given options and check which of the following statements are correct.

Statement (i):

As $0<x<1$, let $x=\dfrac{1}{2}$

$\Rightarrow \dfrac{1}{1 + \frac{1}{2}} < 1 - \dfrac{1}{2} + \dfrac{1}{4}$

$\Rightarrow \dfrac {2}{3} < \dfrac{3}{4}$ which is true.

Therefore, statement (i) is correct.

Statement (ii):

Using the result obtained in statement (i), it is clear that statement (ii) is false.

Statement (iii):

as $-1<x<0$, let $x=-\dfrac{1}{2}$

$\Rightarrow \dfrac{1}{1 - \frac{1}{2}} < 1 - (-\dfrac{1}{2}) + \dfrac{1}{4}$

$\Rightarrow 2 < \dfrac{7}{4}$ which is false.

Statement (iv):

Using the result obtained in statement (iii), it is clear that statement (iv) is true (as $2 > \dfrac{7}{4}$).

Hence, the correct option is $(3)$.