1. **State the problem:** We have an obtuse triangle $\triangle ABC$ with $CB = 20$, $\angle A = 30^\circ$, and $\angle B = 45^\circ$. We want to find the length of side $CA$.\n\n2. **Recall the Law of Sines:** For any triangle, $\frac{\sin \angle A}{a} = \frac{\sin \angle B}{b} = \frac{\sin \angle C}{c}$, where $a, b, c$ are sides opposite angles $A, B, C$ respectively.\n\n3. **Identify sides and angles:** Given $CB = 20$, which is side $a$ opposite $\angle A$. We want $CA = b$, opposite $\angle B$.\n\n4. **Set up the Law of Sines ratio:** $$\frac{\sin \angle A}{a} = \frac{\sin \angle B}{b}$$ Substitute known values: $$\frac{\sin 30^\circ}{20} = \frac{\sin 45^\circ}{b}$$\n\n5. **Solve for $b$:** $$b = \frac{20 \sin 45^\circ}{\sin 30^\circ}$$\n\n6. **Interpretation:** The expression to find $CA$ is: $$\boxed{\frac{20 \sin 45^\circ}{\sin 30^\circ}}$$\n\nThis matches the first expression listed in the problem.