Final answer:
The excluded values for the given product expression are 0, -2, -3, and -8, as setting the denominators of the expression equal to these values would result in division by zero, which is undefined.
Step-by-step explanation:
To find the excluded values for the product (x²-4x-21)/(3x²+6x) * (x²+8x)/(x²+11x+24), we need to consider the values of x for which the denominator is equal to zero, as division by zero is undefined in mathematics. The denominators in this expression are 3x²+6x and x²+11x+24. We set each denominator equal to zero and solve for x:
- 3x² + 6x = 0
- x² + 11x + 24 = 0
For the first denominator, we factor out an x:
x(3x + 6) = 0
Setting each factor equal to zero gives us x = 0 and x = -2 as excluded values. For the second denominator, we can factor it as (x+3)(x+8) = 0, giving us x = -3 and x = -8 as excluded values. Therefore, the excluded values for x are 0, -2, -3, and -8.