Different Bottoms

To add $\frac{1}{2}+\frac{1}{3}$, you first rewrite both over 6: $\frac{3}{6}+\frac{2}{6}$, which is 5/6. The pieces must be the same size before you can count them. The smallest bottom both 2 and 3 go into, 6, is the least common denominator.

Fractions with x work the same way. In $\frac{3}{x}+\frac{1}{x+4}$, the bottoms x and $x+4$ share no factor, so the least common denominator is their product, $x\left(x+4\right)$. Each fraction is multiplied top and bottom by the factor it is missing: the first by $x+4$, the second by x.

$$\frac{3\left(x+4\right)}{x\left(x+4\right)}+\frac{x}{x\left(x+4\right)}$$

Multiplying top and bottom by the same nonzero thing multiplies the fraction by 1, so its value does not change. Now add the tops and collect: $\frac{4x+12}{x\left(x+4\right)}$. Leave the bottom factored, so you can see at a glance that nothing cancels.