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.

If 1/x - 1/y = 4 then (2x + 4xy - 2y)/(x - y + 2xy)

This solution demonstrates how to solve the rational expression (2x + 4xy - 2y)/(x - y + 2xy) using algebraic techniques, including manipulation of the given equation 1/x - 1/y = 4 and factoring. Suitable for high school students in Grades 10-12.

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