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To solve this multiplication of rational expressions, we'll first break down the two fractions, simplify them as much as possible, and then multiply.

We are given:

\[
\frac{x-4}{2x-18} \cdot \frac{6x-54}{7x-28}
\]

### Step 1: Factor each expression.

- For \( 2x - 18 \), factor out the greatest common factor (GCF), which is 2:
  \[
  2x - 18 = 2(x - 9)
  \]
  
- For \( 6x - 54 \), the GCF is 6:
  \[
  6x - 54 = 6(x - 9)
  \]

- For \( 7x - 28 \), the GCF is 7:
  \[
  7x - 28 = 7(x - 4)
  \]

Now we have:
\[
\frac{x-4}{2(x-9)} \cdot \frac{6(x-9)}{7(x-4)}
\]

### Step 2: Simplify.
- We can cancel out the common factors of \(x - 4\) in the numerator and denominator.
- We can also cancel out \(x - 9\) from both the numerator and denominator.

After canceling, we're left with:

\[
\frac{6}{2 \cdot 7}
\]

### Step 3: Multiply and simplify.

\[
\frac{6}{14} = \frac{3}{7}
\]

### Final Answer:
The simplified result of the multiplication is:

\[
\boxed{\frac{3}{7}}
\]

Would you like further details or have any questions about the steps?

Here are five related questions you could explore:
1. How do you factor quadratic expressions in rational equations?
2. What is the process of simplifying complex rational expressions?
3. How do you identify common factors in polynomials?
4. What are some strategies for multiplying and simplifying fractions?
5. What should you do when the numerator or denominator contains a binomial?

**Tip:** Always check for common factors before multiplying fractions to simplify the process!

Multiply: (x - 4) / (2x - 18) * (6x - 54) / (7x - 28) and simplify.

Learn how to multiply and simplify rational expressions like (x-4)/(2x-18) * (6x-54)/(7x-28). This problem involves factoring polynomials, canceling common factors, and simplifying expressions. Suitable for students in Grades 8-10.

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