I will state the problem:
Suggest an algorithm that works in $O(|E| + |V|log|V|)$ time that checks if there are negative cycles in a graph.
So, I saw the runtime, and I immediately said we need to use Dijsktra's implementation with a fibonacci heap. I suggested the following:
Run dijkstra, and mark the distances array as $d$. -- $O(|E| + |V|log|V|)$
Do relax on each edge -- $O(|E|)$
For each edge $<u,v> \in E$: -- $O(|E|)$
3.1 if ( $d[v] < d[u] + w(<u,v>)$), report "negative cycle".
report "No negative cycles"
Would this work?