  Let G be a locally compact non-compact group. We show that under a very mild
assumption on the weight function w, the weighted group algebra L_1(G,w) is
strongly Arens irregular in the sense of Dales-Lamb-Lau. To this end, we first
derive a general factorization theorem for bounded families in the
L_\infty(G,1/w)^*-module L_\infty(G,1/w).
