shyFog36 avatar

To identify the multiplicities of the linear factors of the polynomial \( h(x) = (x + 4)(x - 1)^2(x + 3)^5 \), we look at the exponents of each factor in the expression.

1. The factor \( (x + 4) \) appears once (no exponent), which means its multiplicity is **1**.
2. The factor \( (x - 1)^2 \) has an exponent of 2, so its multiplicity is **2**.
3. The factor \( (x + 3)^5 \) has an exponent of 5, so its multiplicity is **5**.

Thus, the correct identification of the multiplicities is:
- The multiplicity of \( (x + 4) \) is 1,
- The multiplicity of \( (x - 1) \) is 2,
- The multiplicity of \( (x + 3) \) is 5.

Therefore, the correct choice is:
**The multiplicity of \( (x + 4) \) is 1, the multiplicity of \( (x - 1) \) is 2, and the multiplicity of \( (x + 3) \) is 5.**