Step 1: Understand the decay process.
When Uranium-238 (\( ^{238}U \)) undergoes radioactive decay, it transforms into Protactinium-234 (\( ^{234}Pa \)) through a two-step process. The decay of \( ^{238}U \) involves the emission of an \( \alpha \)-particle, followed by the emission of a \( \beta \)-particle.
Step 2: Explanation of emissions.
- The first step involves the emission of an \( \alpha \)-particle, which is a helium nucleus (\( ^4_2He \)), causing a decrease in the atomic number by 2 and the mass number by 4.
- The second step involves the emission of a \( \beta \)-particle, which is an electron, resulting in the conversion of a neutron into a proton.
Step 3: Conclusion.
Thus, the correct emission products in this case are an \( \alpha \)-particle and a \( \beta \)-particle.