We can do this by finding a common denominator for the fractions in the numerator of the function. The common denominator is $4(x+4)$, so we get:
$$\frac{\frac{1}{x+4}-\frac{1}{4}}{x} = \frac{\frac{4-(x+4)}{4(x+4)}}{x} = \frac{\frac{4-x}{4(x+4)}}{x}$$
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