Consider $[x] = L, [m] = M, [t] = T$,
$$ m \frac{d^2x}{dt^2} + \alpha x + \beta x^3 = F_0\sin(\omega t)$$
On the RHS $F_0, \omega$ are also unknown units.
Calculate the dimensions of $\alpha$.
$$\frac{ML}{T^2} + [\alpha]\cdot L + [\beta]\cdot L^3 = [F_0]\sin(\omega t)$$
I know we can't add units of different units, so how can I go about this?
All the terms in your sum should have the same unit.
Consequently $[\alpha]=\dfrac{M}{T^2}$, and I think you can infer the other ones following this example.
Besides, $\sin(\omega t)$ does not have any unit, therefore you can remove it from your last equation. This shows that $[F_0]$ has the same unit as $[\alpha]L$.