Answer
$\frac{x-2}{8}$.
Work Step by Step
Both numerator and denominator of the given expression contain fractions. In order to simplify the given expression we will multiply both numerator and denominator of the given expression by the Least Common Denominator (LCD) of all the fractions inside them.
The LCD of all fractions in the numerator and the denominator is $4x$.
Multiply the numerator and the denominator by LCD.
$\Rightarrow \frac{4x\left (\frac{x}{4}-\frac{1}{x}\right )}{4x\left (1+\frac{x+4}{x}\right )}$
Use the distributive property.
$\Rightarrow \frac{4x\left (\frac{x}{4}\right )-4x\left (\frac{1}{x}\right )}{4x\left (1\right )+4x\left (\frac{x+4}{x}\right )}$
Cancel common factors.
$\Rightarrow \frac{x^2-4}{4x+4(x+4 )}$
Use the distributive property.
$\Rightarrow \frac{x^2-4}{4x+4x+16}$
Add like terms.
$\Rightarrow \frac{x^2-4}{8x+16}$
Factor $x^2-4$.
$\Rightarrow x^2-2^2$
Use the special formula $A^2-B^2=(A+B)(A-B)$.
$\Rightarrow (x+2)(x-2)$
Factor $8x+16$.
Factor out $8$.
$\Rightarrow 8(x+2)$.
Back substitute factors into the fraction.
$\Rightarrow \frac{(x+2)(x-2)}{8(x+2)}$
Cancel common factors.
$\Rightarrow \frac{x-2}{8}$.
