To determine which expression is equivalent to [tex]\(\left(\frac{x^2-25}{x+4}\right) \div\left(\frac{x+5}{x-5}\right)\)[/tex], let's follow these steps.
1. Rewrite the division as multiplication by the reciprocal:
[tex]\[
\left(\frac{x^2-25}{x+4}\right) \div \left(\frac{x+5}{x-5}\right) = \left(\frac{x^2-25}{x+4}\right) \times \left(\frac{x-5}{x+5}\right)
\][/tex]
2. Factor the numerator [tex]\(x^2 - 25\)[/tex]:
[tex]\[
x^2 - 25 = (x + 5)(x - 5)
\][/tex]
This re-writes our expression as:
[tex]\[
\left(\frac{(x+5)(x-5)}{x+4}\right) \times \left(\frac{x-5}{x+5}\right)
\][/tex]
3. Combine the fractions:
[tex]\[
\frac{(x+5)(x-5)}{x+4} \times \frac{x-5}{x+5} = \frac{(x+5)(x-5) \cdot (x-5)}{(x+4)(x+5)}
\][/tex]
4. Simplify the expression by canceling common factors in the numerator and the denominator:
- The factor [tex]\((x+5)\)[/tex] in the numerator and the denominator cancel out.
- The expression now looks like this:
[tex]\[
= \frac{(x-5)^2}{x+4}
\][/tex]
Therefore, the simplified expression is:
[tex]\[
\frac{(x-5)^2}{x+4}
\][/tex]
Upon comparing with the given choices:
A. [tex]\(\frac{x+4}{(x+5)^2}\)[/tex]
B. [tex]\(\frac{x+4}{(x-5)^2}\)[/tex]
C. [tex]\(\frac{(x+5)^2}{x+4}\)[/tex]
D. [tex]\(\frac{(x-5)^2}{x+4}\)[/tex]
The correct choice is:
[tex]\[ \boxed{D} \][/tex]
