Factor the Bottoms First

Take $\frac{2}{{x}^{2}-25}+\frac{1}{x-5}$. Multiplying the bottoms would give a cubic, far more than you need. Factor first: ${x}^{2}-25 = \left(x+5\right)\left(x-5\right)$. Now you can see that the second bottom, $x-5$, is already inside the first.

So the least common denominator is just $\left(x+5\right)\left(x-5\right)$. The first fraction already has it, and the second is only missing $x+5$:

$$\frac{2}{\left(x+5\right)\left(x-5\right)}+\frac{x+5}{\left(x+5\right)\left(x-5\right)}$$

Why factor? A common denominator is built from factors: every factor of every bottom must be in it, and nothing more. Shared factors only show once the bottoms are factored. Adding the tops gives $\frac{x+7}{\left(x+5\right)\left(x-5\right)}$.