Statement -1 : The equation $x^2+(2m+1)x+(2n+1)=0,$ where m, n $\in$ I, cannot have any rational roots.
Statement- 2: The quantity ${(2m+1)}^2-4(2n+1),$ where m, $n\in I,$ can never be perfect square.
Have to select the correct choice as given below.

We have, $x^2+(2m+1)x+(2n+1)=0\mathrm{.....}\left(i\right)$

$m,n\in I$

$\therefore D=b^2-4ac={(2m+1)}^2-4(2n+1)$

Is never be a perfect square.
Therefore, the roots of Eq. (i) can never be integers. Hence, the roots of Eq. (i) cannot have any rational root as $a=1,b,c\in I.$ Hence both statements are true and Statements -2 is a correct explanation of Statement- 1.