Which expression is equivalent to [tex]\left(\frac{x^2-25}{x+4}\right) \div\left(\frac{x+5}{x-5}\right)[/tex]?

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]



Answer :

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]