To find the quotient of the given rational expressions, let's follow these steps:
1. Understand the given problem:
We need to find the quotient of two rational expressions:
[tex]\[
\frac{x^2-16}{x+5} \div \frac{x^2-8x+16}{2x+10}
\][/tex]
2. Rewrite the division as multiplication by the reciprocal:
[tex]\[
\frac{x^2-16}{x+5} \div \frac{x^2-8x+16}{2x+10} = \frac{x^2-16}{x+5} \times \frac{2x+10}{x^2-8x+16}
\][/tex]
3. Factorize the numerators and denominators:
- [tex]\(x^2 - 16\)[/tex] is a difference of squares:
[tex]\[
x^2 - 16 = (x-4)(x+4)
\][/tex]
- [tex]\(x^2 - 8x + 16\)[/tex] is a perfect square trinomial:
[tex]\[
x^2 - 8x + 16 = (x-4)^2
\][/tex]
- [tex]\(2x + 10\)[/tex] can be factored out:
[tex]\[
2x + 10 = 2(x + 5)
\][/tex]
4. Substitute the factored forms into the expressions:
[tex]\[
\frac{(x-4)(x+4)}{x+5} \times \frac{2(x+5)}{(x-4)^2}
\][/tex]
5. Simplify by canceling common factors:
Notice that [tex]\((x+5)\)[/tex] and [tex]\((x-4)\)[/tex] appear in both the numerator and the denominator, so they can be canceled out:
[tex]\[
\frac{(x-4)(x+4)}{\cancel{x+5}} \times \frac{2\cancel{(x+5)}}{(x-4)^2} = \frac{(x-4)(x+4)}{1} \times \frac{2}{(x-4)^2}
\][/tex]
Simplify further:
[tex]\[
\frac{(x+4)}{1} \times \frac{2}{x-4} = \frac{2(x+4)}{x-4}
\][/tex]
Therefore, the quotient of the given rational expressions in reduced form is:
[tex]\[
\boxed{\frac{2(x+4)}{x-4}}
\][/tex]
The correct answer is [tex]\(\boxed{A}\)[/tex].
