The way to crack this is to lock onto the morphology described and let the allergy history confirm it. We are told the cultured material shows numerous hyphae that split into two equal arms at a sharp angle near 45 degrees. That branching pattern, paired with septate filaments, is the textbook signature of Aspergillus species. Layer on the background of asthma, prior allergy and nasal polyps with inspissated secretions, and the whole story reads as allergic fungal rhinosinusitis, a hypersensitivity-driven condition for which Aspergillus is the prototypical culprit. So the pick is option B. The remaining fungi do not fit the picture. Rhizopus and Mucor both belong to the Mucorales and produce wide, ribbon-shaped, aseptate hyphae that branch at near right angles, and they tend to cause angioinvasive mucormycosis in diabetic or immunocompromised hosts rather than allergic polyposis. Candida is a yeast that forms buds and pseudohyphae and does not display the orderly acute-angle dichotomous branching this stem stresses. The angle of branching and the allergic context both converge on the same organism. $B$ is correct. \[\boxed{\text{Aspergillus}}\]