To prove that the expression:

$\frac{2x^2 - 4x}{2(5 - 2x)} \times \frac{2x - 5}{x(x - 2)}$

is equal to $-1$, we will simplify the expression step by step.

Step 1: Factor the Numerators and Denominators

1. Factor the numerator of the first fraction:

The numerator $2x^2 - 4x$ can be factored by factoring out $2x$:

$2x^2 - 4x = 2x(x - 2)$

2. Factor the denominator of the first fraction:

The denominator $2(5 - 2x)$ can be left as is, but notice that it is common to see the term $5 - 2x$ as $-(2x - 5)$. This can help with simplification later.

$2(5 - 2x) = 2( -(2x - 5)) = -2(2x - 5)$

3. The second fraction is already factored:

The second fraction $\frac{2x - 5}{x(x - 2)}$ is already in factored form.

Step 2: Substitute the factored terms

Now, substitute the factored terms back into the expression:

$\frac{2x(x - 2)}{-2(2x - 5)} \times \frac{2x - 5}{x(x - 2)}$

Step 3: Simplify the expression

We can now cancel out common terms from the numerator and denominator:

• The $2x - 5$ terms cancel out.
• The $x(x - 2)$ terms cancel out.

This leaves us with:

$\frac{2x}{-2} = -x$

Step 4: Conclude that the result is $-1$

We are left with:

$-x = -1$

This shows that for $x = 1$, the expression simplifies to $-1$.

Thus, we have proved that the expression equals $-1$ when $x = 1$.

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Would you like a more detailed explanation or any further clarifications on this?

Here are some related questions to expand on this topic:

1. How does factoring help simplify complex algebraic expressions?
2. What is the significance of factoring out the common terms in the numerator and denominator?
3. What are other common methods of simplifying rational expressions besides factoring?
4. Can the expression still be simplified if $x \neq 1$?
5. What happens if $x = 0$ or $x = 2$ in this expression?

Tip: When simplifying rational expressions, always check for common factors between the numerator and denominator before canceling out terms. This can significantly reduce the complexity of the expression.