For a multigraph $G$, $χ'(G)$ denotes the chromatic index of $G$, $Δ(G)$ the maximum degree of $G$, and $Γ(G) = \max\left\{\left\lceil \frac{2|E(H)|}{|V(H)|-1} \right\rceil: H \subseteq G \text{ and } |V(H)| \text{ odd}\right\}$. As a generalization of Vizing's classical coloring result for simple graphs, the Goldberg-Seymour conjecture, posed in the 1970s, states that $χ'(G)=\max\{Δ(G), Γ(G)\}$ or $χ'(G)=\max\{Δ(G) + 1, Γ(G)\}$. Hochbaum, Nishizeki, and Shmoys further conjectured in 1986 that such a coloring can be found in polynomial time. A long proof of the Goldberg-Seymour conjecture was announced in 2019 by Chen, Jing, and Zang, and one case in that proof was eliminated recently by Jing (but the proof is still long); and neither proof has been verified. In this paper, we give a proof of the Goldberg-Seymour conjecture that is significantly shorter and confirm the Hochbaum-Nishizeki-Shmoys conjecture by providing an $O(|V|^5|E|^3)$ time algorithm for finding a $\max\{Δ(G) + 1, Γ(G)\}$-edge-coloring of $G$.