x^2+x >1/e^4 make this an equality and complete the square
x^2 + x + 1/4 = 1/e^4 + 1/4 factor the left side, simplify the right
(x + 1/2)^2 = [ 4 + e^4] / [4e^4) take the square root of both sides
x + 1/2 = +/- sqrt [ 4 + e^4] / [2e^2]
x = +/- sqrt [ 4 + e^4] / [ 2e^2] - 1/2 ≈ [ -1.0179919293664084, 0.0179919293664084 ]
The answer will come from one or more of these intervals
(-infinity, - sqrt [ 4 + e^4] / [ 2e^2] - 1/2) , (- sqrt [ 4 + e^4] / [ 2e^2] - 1/2, sqrt [ 4 + e^4] / [ 2e^2] - 1/2), ( sqrt [ 4 + e^4] / [ 2e^2] - 1/2, infinity)
Choosing 0 as a test point for the middle interval will make the inequality false
So.....the two outside intervals solve the inequality
Here's the graph that proves this........https://www.desmos.com/calculator/iluvf1vudp
x^2+x >1/e^4 make this an equality and complete the square
x^2 + x + 1/4 = 1/e^4 + 1/4 factor the left side, simplify the right
(x + 1/2)^2 = [ 4 + e^4] / [4e^4) take the square root of both sides
x + 1/2 = +/- sqrt [ 4 + e^4] / [2e^2]
x = +/- sqrt [ 4 + e^4] / [ 2e^2] - 1/2 ≈ [ -1.0179919293664084, 0.0179919293664084 ]
The answer will come from one or more of these intervals
(-infinity, - sqrt [ 4 + e^4] / [ 2e^2] - 1/2) , (- sqrt [ 4 + e^4] / [ 2e^2] - 1/2, sqrt [ 4 + e^4] / [ 2e^2] - 1/2), ( sqrt [ 4 + e^4] / [ 2e^2] - 1/2, infinity)
Choosing 0 as a test point for the middle interval will make the inequality false
So.....the two outside intervals solve the inequality
Here's the graph that proves this........https://www.desmos.com/calculator/iluvf1vudp