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.