Number the enone carbonyl carbon C1, the adjacent alkene carbon C2 and the ethoxy-bearing alkene carbon C3. In the bromine-first pathway, the enol-ether-like unit attacks at C2, breaking the Br–Br bond. The intermediate has a new C2–Br bond and positive charge at C3, stabilized as an oxonium chemical resonance contributor by donation from the ethoxy oxygen.
triethylamine removes the remaining C2 hydrogen. Its C–H bonding pair reforms C2=C3 and neutralizes the oxonium system. The result is replacement of the vinylic C2–H by C2–Br, with the original enone and ethoxy group retained. If bromide first captures the cationic intermediate, the equivalent vicinal dibromide can eliminate HBr to reach the same product.
The ethoxy group makes this alkene sufficiently electron-rich for the electrophilic-addition description. It is important to place bromine at C2 and restore the double bond, rather than stop at a saturated addition product.
For an amine-first pathway, use triethylamine as a nucleophile in a reversible conjugate addition. Its lone pair attacks C3 of the enone; electron shifts through C2 and the carbonyl give a zwitterionic enolate with a C3– bond.
The C2 nucleophilic site of that enolate attacks , installing C2–Br and releasing bromide. Removal of C2–H by base then reforms C2=C3 while the C3–N bond breaks back to neutral triethylamine. This addition–bromination–elimination sequence gives the same C2-brominated enone and regenerates the nucleophilic amine, with another base equivalent accepting the eliminated proton.
These are alternative formal sequences requested by the question, not a kinetic determination that one must dominate under unspecified conditions. In particular, the amine-first sequence does not require triethylamine to generate a high concentration of a vinylic carbanion by direct deprotonation.
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