Final answer:
The zeros of the function h(x) = 3x⁴ - 16x³ + 38x² - 56x + 15 are 1 with a multiplicity of 3 and 5 with a multiplicity of 1.
Step-by-step explanation:
To identify the zeros and their multiplicities of the function h(x) = 3x⁴ - 16x³ + 38x² - 56x + 15, we need to factor the function and analyze the factors.
By using synthetic division or the long division method, we find that the function can be factored as follows: (x-1)(x-1)(x-1)(x-5).
Therefore, the zeros of h(x) are 1 with a multiplicity of 3 and 5 with a multiplicity of 1. So, option a) Zeros: 1 with multiplicity 3, 5 with multiplicity 1 is correct.