Find the greatest common factor, GCF, of the group of expressions \[ 10(x+3)(x-4)^{2} \text { and } 6(x-4)^{3}(x-3) \]

To find the greatest common factor (GCF) of the expressions \( 10(x+3)(x-4)^{2} \) and \( 6(x-4)^{3}(x-3) \), we need to first identify the GCF of the coefficients and then the GCF of the variable factors. **Step 1: Finding GCF of the coefficients** - The coefficients are 10 and 6. - The factors of 10 are \( 1, 2, 5, 10 \). - The factors of 6 are \( 1, 2, 3, 6 \). - The GCF of 10 and 6 is \( 2 \). **Step 2: Finding GCF of the variable factors** - For the variable \( (x+3) \), it appears only in the first expression, thus it contributes nothing to the GCF. - For the variable \( (x-4) \), the powers are \( 2 \) in the first expression and \( 3 \) in the second expression. The GCF is \( (x-4)^{2} \) (the lower power). - For the variable \( (x-3) \), it appears only in the second expression, thus it also contributes nothing to the GCF. **Step 3: Combining the results** Now, we combine the GCF of the coefficients and the variable factors: \[ \text{GCF} = 2 \cdot (x-4)^{2} \] Thus, the greatest common factor of the group of expressions is: \[ \boxed{2(x-4)^{2}} \]