Perform the following operation and express in simplest form.

start fraction, x, squared, minus, 6, x, minus, 16, divided by, x, end fraction, dot, start fraction, 2, x, divided by, x, squared, minus, 4, end fraction

*xx*2−6*x*−16​⋅*x*2−42*x*​


We are tasked with simplifying the following expression:

\[
\frac{x^2 - 6x - 16}{x} \cdot \frac{2x}{x^2 - 4}
\]

### Step 1: Factor both expressions

1. **Factor the quadratic expressions:**

- \(x^2 - 6x - 16\) can be factored as \((x - 8)(x + 2)\).
- \(x^2 - 4\) is a difference of squares and can be factored as \((x - 2)(x + 2)\).

### Step 2: Rewrite the expression

After factoring, the expression becomes:

\[
\frac{(x - 8)(x + 2)}{x} \cdot \frac{2x}{(x - 2)(x + 2)}
\]

### Step 3: Simplify the expression

We can now cancel common factors:

- The \(x + 2\) terms cancel out from the numerator and denominator.
- The \(x\) in the numerator of the second fraction cancels with the \(x\) in the denominator of the first fraction.

The simplified expression is:

\[
\frac{2(x - 8)}{x - 2}
\]

Thus, the simplified form of the expression is:

\[
\frac{2(x - 8)}{x - 2}
\]

### Final Answer:

\[
\frac{2(x - 8)}{x - 2}
\]

Would you like to explore further details or ask additional questions about this process?

Here are 5 related questions for further practice:
1. How would you simplify \(\frac{x^2 - 25}{x + 5} \cdot \frac{x + 5}{x - 5}\)?
2. Can you factor \(x^2 + 7x + 12\)?
3. What happens when you multiply two rational expressions where both have the same denominator?
4. How would you simplify \(\frac{x^2 - 1}{x + 1} \cdot \frac{x + 1}{x - 1}\)?
5. How do you simplify \(\frac{x^2 + 5x}{x^2 - x}\)?

**Tip:** Always factor both the numerator and denominator before attempting to simplify rational expressions.

Perform the following operation and express in simplest form. (x^2 - 6x - 16) / x * (2x / (x^2 - 4))

Learn how to simplify the expression (x^2 - 6x - 16) / x * (2x / (x^2 - 4)) by factoring and canceling common terms. This step-by-step solution covers key algebra concepts, including factoring quadratics and simplifying rational expressions, ideal for students in Grades 9-10.

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