The internal energy (U), pressure (P) and volume (V) of an ideal gas are related as $$U = 3PV + 4$$. The gas is

For an ideal gas, the internal energy is related to pressure and volume by $$U = \frac{f}{2}nRT = \frac{f}{2}PV$$, where $$f$$ is the number of degrees of freedom.

We are given $$U = 3PV + 4$$. The constant 4 does not affect the relationship between changes in $$U$$ and $$PV$$, so the effective relationship is $$U = 3PV + \text{constant}$$, meaning $$\frac{f}{2} = 3$$, giving $$f = 6$$.

For a monoatomic gas, $$f = 3$$. For a diatomic gas (at moderate temperatures), $$f = 5$$. For a polyatomic gas, $$f = 6$$ (3 translational + 3 rotational degrees of freedom).

Since $$f = 6$$, the gas is polyatomic only.