To determine the restrictions for the expression [tex]\(\frac{5x + 10}{x^2 + 10x + 24} \times \frac{x + 8}{x^2 - 16}\)[/tex], we need to find where the denominators are zero because division by zero is undefined.
Let's break down the problem step-by-step:
1. Identify the denominators:
- The first denominator is [tex]\(x^2 + 10x + 24\)[/tex].
- The second denominator is [tex]\(x^2 - 16\)[/tex].
2. Factor the first denominator [tex]\(x^2 + 10x + 24\)[/tex]:
- We need to find factors of 24 that add up to 10. These factors are 6 and 4.
- Thus, [tex]\(x^2 + 10x + 24 = (x + 6)(x + 4)\)[/tex].
3. Factor the second denominator [tex]\(x^2 - 16\)[/tex]:
- This is a difference of squares, so it factors as [tex]\(x^2 - 16 = (x + 4)(x - 4)\)[/tex].
4. Find the values of [tex]\(x\)[/tex] that make the denominators zero:
- From the first denominator: [tex]\((x + 6)(x + 4) = 0 \implies x = -6 \text{ or } x = -4\)[/tex].
- From the second denominator: [tex]\((x + 4)(x - 4) = 0 \implies x = -4 \text{ or } x = 4\)[/tex].
5. Collect all the restrictions:
- Combine the values from both factorizations: [tex]\(x = -6, -4, 4\)[/tex].
Therefore, the restrictions for the given expression are that [tex]\(x\)[/tex] cannot be -6, -4, or 4.
Comparing these with the given options, the restrictions are:
- [tex]\(x \neq -4\)[/tex]
- [tex]\(x \neq 4\)[/tex]
- [tex]\(x \neq -6\)[/tex]
So, the correct choices given the options are:
- [tex]\(x \neq -4, 4\)[/tex]
- [tex]\(x \neq -6\)[/tex]
