Answer
$ f $ cannot have any positive real zeroes and cannot have any negative real zeroes.
Work Step by Step
Consider a polynomial $ f(x)=a_nx^n+a_{n-1} x^{n-1} +....+a_2 x^2+a_1x+a_0$ with real coefficients and $ a_0 \ne 0$
Remember: 1) The number of positive real zeros of $ f(x)$ is less than or equal to the number of variations in sign of $ f(x)$.
2)The number of negative real zeros of $ f(x)$ is less than or equal to the number of variations in sign of $ f(-x)$.
We see that $ f(x)=f(-x)=2x^4+6x^2+8$
Thus, the function has no real roots.
