The problem asks us to find the greatest common factor (GCF) of the two expressions: $15(x+1)^3(x-1)^4$ and $10(x+1)^5(x-1)$. We must select the correct answer among the provided options.

1. Problem Description

The problem asks us to find the greatest common factor (GCF) of the two expressions: $15(x+1)^3(x-1)^4$ and $10(x+1)^5(x-1)$. We must select the correct answer among the provided options.

2. Solution Steps

To find the GCF of two expressions, we find the GCF of the coefficients and the lowest power of each common factor.

The two expressions are $15(x+1)^3(x-1)^4$ and $10(x+1)^5(x-1)$.

First, find the GCF of the coefficients 15 and

1

0. The prime factorization of 15 is $3 \cdot 5$, and the prime factorization of 10 is $2 \cdot 5$. The GCF of 15 and 10 is

5.

Next, we find the lowest power of the common factor $(x+1)$.

The powers of $(x+1)$ are 3 and

5. The lowest power is 3, so the GCF contains $(x+1)^3$.

Then, we find the lowest power of the common factor $(x-1)$.

The powers of $(x-1)$ are 4 and

1. The lowest power is 1, so the GCF contains $(x-1)^1$ or simply $(x-1)$.

Multiplying these together, the GCF is $5(x+1)^3(x-1)$.

3. Final Answer

The greatest common factor is $5(x+1)^3(x-1)$. This corresponds to option (b).