If complex coefficients are allowed, the answer is 3.
If the polynomial must have real coefficients, then each complex root comes as a pair of complex conjugate roots.
Root -5 is real, so that is 1 root, and degree 1.
Root 1 + 4i is complex, so it must come with its complex conjugate, 1 - 4i. This adds 2 roots to the polynomial, and now we're up to degree 3.
Root -4i is also complex. It also must come with its complex conjugate, 4i. That adds two more roots, and the degree is 5.
Answer: The least possible degree is 5 with real coefficients.
