When is the inequality x^4 < 8x^2 + 9 true?

If we consider this like a normal quadratic problem, this becomes easy

x⁴ < 8x² + 9

x⁴ - 8x² - 9 < 0

(x²-9)(x²+1) < 0

This means there are roots of this expression at x² = 9 and x² = -1

Since for all reals, x² > 0, we know the two roots of this expression are x=+-3

Now, since x⁴ - 8x² - 9 is a quartic (ie, it has an x⁴ expression), we know that given any sufficiently positive or negative x, the quartic will be positive (ie, if x is 10000, or -10000)

Therefore, we know for this to be true, -3<x<3 (since we have found the solutions, we simply need to work out which regions satisfy the criteria)