We are given the equation:

$\frac{1}{x} - \frac{1}{y} = 4$

First, we can rewrite this equation as:

$\frac{y - x}{xy} = 4$

Multiplying both sides by $xy$, we obtain:

$y - x = 4xy$

Now, let's evaluate the given expression:

$\frac{2x + 4xy - 2y}{x - y + 2xy}$

We know that $x - y = -4xy$, so substitute this into the expression:

$\frac{2x + 4xy - 2y}{-4xy + 2xy}$

Simplify the denominator:

$-4xy + 2xy = -2xy$

Now substitute this back into the fraction:

$\frac{2x + 4xy - 2y}{-2xy}$

We can simplify the numerator $2x + 4xy - 2y$ by factoring it:

$2(x - y + 2xy)$

Thus, the expression becomes:

$\frac{2(x - y + 2xy)}{-2xy}$

Since $x - y + 2xy = 0$ (from $x - y = -4xy$), we get:

$\frac{2(0)}{-2xy} = 0$

Hence, the value of the expression is:

$\boxed{0}$

Would you like further details or explanations?

Related Questions:

1. What happens if $x$ and $y$ are functions rather than constants?
2. How does changing the equation $\frac{1}{x} - \frac{1}{y} = 4$ affect the solution?
3. Can this result be generalized for other values on the right-hand side of the equation?
4. What are the implications if $xy = 0$ for the given expression?
5. How does factoring influence the simplification of algebraic expressions?

Tip:

When simplifying complex fractions, look for opportunities to factor both the numerator and the denominator to reduce the expression efficiently.