A polynomial is an expression where the exponents of the variables are non-negative integers. The given expression is $f(x) = \frac{3}{x^2} + x$. The term $\frac{3}{x^2}$ can be rewritten as $3x^{-2}$. In this term, the exponent of $x$ is $-2$, which is a negative integer. Since a polynomial requires all variable exponents to be non-negative integers, $f(x)$ is not a polynomial.