https://file.aiquickdraw.com/mathbot/file/1729323896984bx5uzxzm.jpg

The expression in the image is as follows:

\[
\frac{4x + 4y}{x - 3y} \cdot \frac{x^2 - 9y^2}{x + y}
\]

Let's simplify it step by step.

### Step 1: Factor where possible
We can start by factoring the second fraction. Notice that the numerator \(x^2 - 9y^2\) is a difference of squares:
\[
x^2 - 9y^2 = (x - 3y)(x + 3y)
\]
Now the expression becomes:
\[
\frac{4x + 4y}{x - 3y} \cdot \frac{(x - 3y)(x + 3y)}{x + y}
\]

### Step 2: Simplify common terms
We can cancel the \(x - 3y\) term from the numerator and the denominator:
\[
\frac{4x + 4y}{1} \cdot \frac{x + 3y}{x + y}
\]

### Step 3: Factor out the common factor in the first fraction
In the numerator \(4x + 4y\), factor out 4:
\[
4(x + y)
\]
Now the expression becomes:
\[
\frac{4(x + y)}{1} \cdot \frac{x + 3y}{x + y}
\]

### Step 4: Cancel common terms
Cancel the \(x + y\) term from the numerator and the denominator:
\[
4 \cdot (x + 3y) = 4(x + 3y)
\]

### Final Answer:
\[
4(x + 3y)
\]

Let me know if you'd like more details or if you have any questions.

### Relative Questions:
1. How do you factor a difference of squares?
2. Why do we cancel terms when simplifying expressions?
3. Can all expressions be factored and simplified?
4. What are the restrictions on \(x\) and \(y\) in this problem to avoid division by zero?
5. How does factoring simplify multiplication of fractions?

### Tip:
Always check for common factors in the numerator and denominator to simplify expressions quickly!

Simplify the expression: (4x + 4y) / (x - 3y) * (x^2 - 9y^2) / (x + y)

Learn how to simplify the rational expression (4x + 4y)/(x - 3y) * (x^2 - 9y^2)/(x + y) using factoring techniques and algebraic simplification. This step-by-step solution covers algebra concepts such as factoring differences of squares and simplifying common terms.

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