For what real values of x is -4< x^4+4x^2< 21? express your answer in interval notation.!

The expression in x is always greater than or equal to zero (because x to an even power is never negative). Hence the limit -4 is irrelevant and we only need to consider the limit 21.

After subtracting 21, the expression can be factored as

... x⁴ +4x² -21 < 0

... (x²+7)(x²-3) < 0

... (x²+7)(x+√3)(x-√3) < 0 . . . . . factoring the difference of squares

The left factor is always positive, so the expression will be negative when one, but not both, of the factors involving √3 is negative. That happens when

... -√3 < x < √3

In interval notation: (-√3, √3).

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