AITS

According to question,

e-ln(2m^2+m+1) < e-ln(3m^2 -4m+1)

=> 1/(2m2+m+1) < 1/(3m2-4m+1)...........................(1)
Cross multiplying both sides of (1) by 2m2+m+1 without changing the direction of the inequality, {Since 2m2+m+1 is always positive on account of Discriminant <0} we have,
(2m2+m+1)/(3m2-4m+1) -1 > 0
=> (-m2+5m)/(3m2-4m+1) > 0
=> m(m-5)/(3m2-4m+1) < 0 {Since the inequality has been multiplied by -1, sign is reversed} => m(m-5)/(3m-1)(m-1) < 0 .....................(2)
And now, the solution of (2) gives the range of m:-
m belongs to (0,1/3) U (1,5)

http://www.goiit.com/posts/list/differential-calculus-if-f-x-e-ln-x-and-f-2m-2-m-1-f-3m-2-4m-1006749.htm