Find the values of 'a' for which `-3<(x^2+ax-2)/(x^2+x+1)<2` is valid for all real `x.`

Explanation

The system is equivalent to
`{((x^(2)-(a+2)x+4)/(x^(2)-x+1)gt0),((4x^(2)+(a-3)x+1)/(x^(2)-x+1)gt0):}`
since `x^(2)-x+1=(x-1/2)^(2)+3/4gt0`, this system is
equivalent to `{(x^(2)-(a+2)x+4gt0),(4x^(2)+(a-3)x+1gt0):}`
Hence the discriminants of the both equations of this system are negative.
i.e. `{((a+2)^(2)-16lt0),((a-3)^(2)-16lt0):}implies(a+6)(a-2)lt0`

i.e. `x epsilon(-1,7)`
Hence, from Eqs i and ii we get
`x epsilon(-1,2)`