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The given task asks to find the value of the expression:

\[
\frac{2x - y}{2x + y}
\]

given the equation:

\[
\frac{4x^2 - 4xy + 3y^2}{2y^2 + 2xy - 5x^2} = 1.
\]

### Step 1: Analyze the given equation
The equation provided is:

\[
\frac{4x^2 - 4xy + 3y^2}{2y^2 + 2xy - 5x^2} = 1.
\]

To simplify, multiply both sides of the equation by the denominator:

\[
4x^2 - 4xy + 3y^2 = 2y^2 + 2xy - 5x^2.
\]

### Step 2: Rearrange the terms
Now, move all terms to one side:

\[
4x^2 - 4xy + 3y^2 - 2y^2 - 2xy + 5x^2 = 0.
\]

Simplify:

\[
(4x^2 + 5x^2) + (-4xy - 2xy) + (3y^2 - 2y^2) = 0,
\]

which simplifies further to:

\[
9x^2 - 6xy + y^2 = 0.
\]

### Step 3: Factor the equation
We can attempt factoring the expression \(9x^2 - 6xy + y^2\). Notice that this is a perfect square trinomial:

\[
(3x - y)^2 = 0.
\]

Thus, we have:

\[
3x - y = 0,
\]

which simplifies to:

\[
y = 3x.
\]

### Step 4: Substitute into the required expression
Now, substitute \(y = 3x\) into the original expression \(\frac{2x - y}{2x + y}\):

\[
\frac{2x - 3x}{2x + 3x} = \frac{-x}{5x} = -\frac{1}{5}.
\]

### Final Answer:
The value of the expression is \(-\frac{1}{5}\).

Would you like further details or have any questions?

Here are 5 questions for further exploration:
1. How does factoring help solve quadratic equations like in this example?
2. What other methods could be used to solve \(9x^2 - 6xy + y^2 = 0\)?
3. How can this process be generalized for more complex expressions?
4. Why is it important to check whether a trinomial is a perfect square?
5. Can you think of similar problems involving substitution techniques?

**Tip:** Always check if an expression can be factored into a perfect square to simplify solving equations.

Найдите значение выражения (2x - y) / (2x + y), если известно, что (4x^2 - 4xy + 3y^2) / (2y^2 + 2xy - 5x^2) = 1.

Learn how to solve the expression (2x - y) / (2x + y) by simplifying and factoring the given equation (4x^2 - 4xy + 3y^2) / (2y^2 + 2xy - 5x^2) = 1. This detailed step-by-step solution covers important algebraic concepts like factoring perfect squares and solving through substitution. Suitable for Grades 9-11.

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