The given expression is:

\[
\frac{49x^2 - 81y^2}{14x + 18y}
\]

We can approach this by factoring the numerator, which appears to be a difference of squares.

### Step 1: Factor the numerator

The numerator is of the form \( a^2 - b^2 \), which can be factored as \( (a - b)(a + b) \). Here, we have:

\[
49x^2 - 81y^2 = (7x)^2 - (9y)^2
\]

Thus, the factorization is:

\[
(7x - 9y)(7x + 9y)
\]

### Step 2: Rewrite the expression

Now that we've factored the numerator, the expression becomes:

\[
\frac{(7x - 9y)(7x + 9y)}{14x + 18y}
\]

### Step 3: Factor the denominator

Factor out the greatest common factor (GCF) from the denominator:

\[
14x + 18y = 2(7x + 9y)
\]

### Step 4: Simplify the expression

Now, the expression looks like this:

\[
\frac{(7x - 9y)(7x + 9y)}{2(7x + 9y)}
\]

We can cancel out \( (7x + 9y) \) from both the numerator and denominator, assuming \( 7x + 9y \neq 0 \) (otherwise, the expression would be undefined).

Thus, the simplified expression is:

\[
\frac{7x - 9y}{2}
\]

### Final Answer:

The simplified form of the given expression is:

\[
\frac{7x - 9y}{2}
\]

Do you need more details or have any questions? 

Here are 5 related questions:
1. How do you factor differences of squares in other expressions?
2. What happens when \(7x + 9y = 0\)?
3. Can you apply the same factoring technique for higher-degree polynomials?
4. How do you handle expressions with more complex denominators?
5. What if the numerator and denominator shared no common factors?

**Tip:** Always check for the greatest common factor (GCF) before simplifying any algebraic expression.

Learn how to simplify the algebraic expression 49x² - 81y² / 14x + 18y step-by-step by factoring using the difference of squares and the greatest common factor. This solution covers key concepts such as factoring, simplification, and rational expressions. Suitable for students in Grades 9-11.

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