Answer
False.
Work Step by Step
This is false. For instance, if we use $x=1$, we get:
Left side: $\displaystyle \frac{1}{x+4}=\frac{1}{1+4}=\frac{1}{5}$
Right side: $\displaystyle \frac{1}{x}+\frac{1}{4}=\frac{1}{1}+\frac{1}{4}=\frac{4}{4}+\frac{1}{4}=\frac{5}{4}$
And the sides do not equal.