Using derivatives we can show that this even function is decreasing on $(-\infty,0)$ and increasing on $[0,\infty)$ has an absolute minima in $(0,-\frac32)$ . Also the function is zero only in $-1$ an $1$, so the solution of the inequation is $[-1,1]$
However, I wonder if there is an elementary (algebraic) solution for the problem.
Obviously $f(x) = 2^{x^4}+2^{x^2-1} $ is even and it is strictly increasing for $x\geq 0$. (We use the fact that composition of two increasing functions is increasing function and a sum of two increasing functions is also increasing function.)
It is easy to guess that $x=1$ is a solution to this inequality. So each $x\in[0,1]$ is a solution. Since $f$ is even we have $x\in[-1,1]$.