To solve this problem, let us simplify and evaluate the given expression step-by-step:

Given:

1. $x - y = 5$
2. Expression to evaluate: $x^2 + y^2 - 4x + 4y - 2xy - 8$

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Step 1: Expand and rearrange the terms

Group the terms logically: $x^2 + y^2 - 2xy - 4x + 4y - 8$

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Step 2: Use the identity for $x^2 + y^2 - 2xy$

Recall the identity: $x^2 + y^2 - 2xy = (x - y)^2$ Substitute $x - y = 5$: $x^2 + y^2 - 2xy = (x - y)^2 = 5^2 = 25$

Now the expression becomes: $25 - 4x + 4y - 8$

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Step 3: Simplify further

Combine constants: $25 - 8 = 17$ Thus: $17 - 4x + 4y$

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Step 4: Express $y$ in terms of $x$

From $x - y = 5$, solve for $y$: $y = x - 5$

Substitute $y = x - 5$ into the remaining expression: $17 - 4x + 4(x - 5)$

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Step 5: Simplify

Expand and simplify: $17 - 4x + 4x - 20$ $17 - 20 = -3$

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Final Answer:

The value of the expression is: $\boxed{-3}$

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Would you like a detailed explanation of any specific step or further questions? 😊

Related Questions:

1. What happens if $x - y \neq 5$ but another linear equation is given?
2. Can you generalize this solution for $x - y = k$ (where $k$ is any constant)?
3. How does the identity $x^2 + y^2 - 2xy = (x - y)^2$ simplify algebraic expressions in general?
4. How can this problem be solved graphically by interpreting $x - y = 5$?
5. What is the geometric interpretation of $x^2 + y^2 - 2xy$?

Tip: Always look for algebraic identities like $a^2 + b^2 - 2ab = (a - b)^2$ to simplify quadratic expressions efficiently!