Dividing

Dividing by a fraction is multiplying by its reciprocal, the fraction turned upside down: $\frac{\frac{2}{3}}{\frac{4}{9}}$ is $\frac{2}{3}\cdot \frac{9}{4}$, which is 3/2. The reason: a fraction times its reciprocal is 1, so multiplying by $\frac{9}{4}$ undoes multiplying by $\frac{4}{9}$, which is exactly what dividing by it does.

Rational expressions follow the same rule. In $\frac{\frac{x+1}{x+5}}{\frac{x+1}{{x}^{2}-25}}$, keep the first fraction and flip the second: $\frac{x+1}{x+5}\cdot \frac{{x}^{2}-25}{x+1}$. Now it is a multiplication. Factor ${x}^{2}-25$ as $\left(x+5\right)\left(x-5\right)$, multiply across, and cancel x + 1 and x + 5. What is left is $x-5$.

Only the second fraction flips. The excluded values grow here: 5 and -5 from the bottoms, and also -1, because at -1 the fraction you divide by is 0, and nothing can be divided by 0.