Find all real value of k in the given inequality.

Question

Find all the real values of k for which the following system of inequalities
$-3<{x^2+kx-2\over x^2 -x +1} < 2$
is fullfilled for all real values of $x$

I have attempted this question but my teacher told me that we have to take $\Delta <0$ in $-3(x^2-x+1)< x^2 + kx -2$

The answer is to be provided in intervals

Answer 1

Continue where you left off: $4x^2+(k-3)x + 1 > 0, \forall x\in \mathbb{R}\implies \triangle_1 < 0\implies (k-3)^2-16<0\implies (k-7)(k+1) < 0\implies -1 < k < 7$, and similarly the other one gives: $\triangle_2 = (k+2)^2 - 16 < 0\implies (k+6)(k-2) < 0\implies -6 < k < 2$. Combine the two inequalities, you have: $-1 < k < 2$