You have everything correct so far
Slopes of the asymptotes : ± b/a = ±6/8 = ±3/4
Equations of the asymptotes : y + 2 = ±(3/4)(x + 5)
Neither of the choices for the equations of the asymptotes are correct
See the graph here : https://www.desmos.com/calculator/aetqrwxf6k