Find the values of `k` for which `|(x^2+k x+1)/(x^2+x+1)|<2,AAx in R`

Explanation

We have
`|x| lt a `rArr - a lt x lt + a`
Therefore, the given inequality implies
`-2lt(x^(2) + kx + 1)/(x^(2) + x + 1) lt 2` (2)
Now, `x^(2) + x +1 = (x+ 1//x)^(2) + (3//4)` is positive for all values of x.
Multiplying (1) by `x^(2) + x + 1,` we get
` -2(x^(2) + x+1) lt x^(2) + kx + 1 lt 2(x^(2) + x + 1)`
This yields two inequalities, viz.,
`3x^(2) + (2+k)x + 3) gt`0
and `x^(2) + (2-k) x + 1 lt 0`
For above inequalities to be. ture for all values of x, their discrimainats must be negative . Hence,
`(2+k)^(2) - 36 lt 0 and (2 - k)^(2) - 4 lt 0` (2)
`rArr (k+ 8) (k - 4) lt 0 and k (k - 4) lt 0` (3)
`rArr - 8 lt k lt 4 and 0 lt k lt 4`
Therefore, `0lt k lt 4`