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.