The correct option is B (−∞,−454)
(x2+x+1)2−(m−3)(x2+x+1)+m=0
Let t=x2+x+1=(x+12)2+34
⇒t∈[34,∞)
Given equation becomes t2−(m−3)t+m=0 ⋯(1)
Let its roots be t1 and t2
(i) For every t>34, there exist two distinct real roots for x2+x+1=t
(ii) For every t<34, there exists no real roots for x2+x+1=t
Given equation will have two distinct roots iff for equation (1), roots are of form
t1<34 and t2>34
i.e., 34 lies in between the roots of equation (1).
⇒f(34)<0
⇒916−(m−3)34+m<0
⇒m<−454