What is the least possible degree of a polynomial that has roots -5, 1+4i, and -4i

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.

The least possible degree is 5 with real coefficients.