We are tasked with simplifying the following rational expression:

$\frac{x^2 - 3x - 54}{x^2 - 9x} \cdot \frac{2x}{x^2 + 8x + 12}.$

Step-by-step Solution:

1. Factorize each part of the expression:

   • Factorize the numerator $x^2 - 3x - 54$: $x^2 - 3x - 54 = (x - 9)(x + 6).$

   • Factorize the denominator $x^2 - 9x$: $x^2 - 9x = x(x - 9).$

   • Factorize the denominator $x^2 + 8x + 12$: $x^2 + 8x + 12 = (x + 6)(x + 2).$

   Substituting these factorizations, the expression becomes: $\frac{(x - 9)(x + 6)}{x(x - 9)} \cdot \frac{2x}{(x + 6)(x + 2)}.$

2. Simplify by canceling common factors:

   • $x - 9$ appears in both the numerator and denominator, so cancel $x - 9$.
   • $x + 6$ appears in both the numerator and denominator, so cancel $x + 6$.
   • $x$ in the numerator of the second fraction cancels with the $x$ in the denominator of the first fraction.

   After cancellation, we are left with: $\frac{2}{x + 2}.$

Final Answer:

$\frac{2}{x + 2}.$

Note:

The expression is valid for $x \neq 9, -6, 0, -2$, as these values would make the original denominators equal to zero.

Do you want further clarification or details on any of these steps?

Related Questions:

1. How do you factor trinomials like $x^2 - 3x - 54$?
2. Why is it important to find restrictions on $x$ in rational expressions?
3. What happens if a factor is not canceled correctly in such problems?
4. How can you confirm your factorizations are correct?
5. How do you approach similar problems involving division instead of multiplication?

Tip:

Always check for factors common to both the numerator and denominator before simplifying—it helps avoid mistakes!