\frac{2-x}{\:x} is this the same as this \frac{4-x^2}{x\left(x+2\right)}

We are given two expressions:

1. \( \frac{2 - x}{x} \)
2. \( \frac{4 - x^2}{x(x + 2)} \)

Let's check if they are equivalent by simplifying the second expression and comparing it to the first.

### Step 1: Factor \( 4 - x^2 \)

The expression \( 4 - x^2 \) is a difference of squares, and it can be factored as:

\[
4 - x^2 = (2 - x)(2 + x)
\]

So the second expression becomes:

\[
\frac{(2 - x)(2 + x)}{x(x + 2)}
\]

### Step 2: Simplify the second expression

Now, cancel out common factors. Notice that \( x + 2 = 2 + x \), so we can cancel \( 2 + x \) from the numerator and denominator:

\[
\frac{(2 - x)(2 + x)}{x(x + 2)} = \frac{2 - x}{x}
\]

### Conclusion:

Both expressions simplify to \( \frac{2 - x}{x} \), so they are indeed the same.

---

Do you want further details on this? 

Here are five related questions:
1. How do you factor a difference of squares in general?
2. What other types of common factorizations are there?
3. How do you know when to cancel terms in a rational expression?
4. Can this equivalence be represented graphically on a coordinate plane?
5. How would this approach change if we had a cubic expression?

**Tip**: Always look for opportunities to factor expressions before simplifying them.

Is \( \frac{2 - x}{x} \) the same as \( \frac{4 - x^2}{x(x + 2)} \)?

Understand how to simplify the rational expression \( \frac{4 - x^2}{x(x + 2)} \) and compare it with \( \frac{2 - x}{x} \) using the difference of squares factorization. This problem covers key algebraic concepts like factoring and rational expressions, suitable for students in Grades 9-12.

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