The inequality \[x^4<8x^2+9\] is satisfied precisely when \[\text{(a)}-3<x<3 \quad \text{(b)} \quad 0<x<4 \quad \text{(c)} \quad 1<x<3 \quad \text{(d)}-1<x<9 \quad \text{(e)}-3<x<-1.\]
Now \(x^2 + 1\) is always positive, so we are interested in when \((x+3)(x-3)\) is negative.
When \(x < -3\), both brackets are negative, and when \(x > 3\), both brackets are positive.
In between these values, one bracket is negative and one is positive, and so \((x+3)(x-3)\) is negative.
Thus the answer is (a).
