Explanation

Note: the stem is truncated in the source after 'such that 1'; the standard completion is 'such that 1<α<2 and 2<β<3 — how many integer values can λ take?' Dividing by 4 gives x²−4x+λ/16 = 0, so the roots are x = 2±√(4−λ/16). With d = √(4−λ/16) ≥ 0, the inequalities 1<2−d and 2+d<3 force 0<d<1, i.e. 0<4−λ/16<1. This yields 48<λ<64, so λ ∈ {49,…,63}: 15 values (option D).