https://file.aiquickdraw.com/mathbot/file/1725922368335yvm1kn9l.jpg

Let's work through the given problem step by step. The goal is to factor the expressions and find the simplified product of these rational expressions.

### Expressions:

\[
\frac{x^2 - 49}{x^2 + 9x + 20} \times \frac{x^2 - 25}{x^2 - 6x - 7}
\]

### Step 1: Factor each expression

#### Numerator \(x^2 - 49\) (Difference of squares):
\[
x^2 - 49 = (x + 7)(x - 7)
\]

#### Denominator \(x^2 + 9x + 20\) (Factorable quadratic):
We need to find two numbers that multiply to 20 and add to 9. These numbers are 5 and 4.
\[
x^2 + 9x + 20 = (x + 5)(x + 4)
\]

#### Numerator \(x^2 - 25\) (Another difference of squares):
\[
x^2 - 25 = (x + 5)(x - 5)
\]

#### Denominator \(x^2 - 6x - 7\) (Factorable quadratic):
We need two numbers that multiply to -7 and add to -6. These numbers are -7 and 1.
\[
x^2 - 6x - 7 = (x - 7)(x + 1)
\]

### Step 2: Substitute the factored forms into the original expression

\[
\frac{(x + 7)(x - 7)}{(x + 5)(x + 4)} \times \frac{(x + 5)(x - 5)}{(x - 7)(x + 1)}
\]

### Step 3: Cancel common factors

- The factor \( (x + 5) \) appears in both the numerator and the denominator, so it cancels out.
- The factor \( (x - 7) \) appears in both the numerator and the denominator, so it cancels out.

This leaves us with:

\[
\frac{(x + 7)(x - 5)}{(x + 4)(x + 1)}
\]

### Final Answer:
The simplified product is:

\[
\frac{(x + 7)(x - 5)}{(x + 4)(x + 1)}
\]

Now, in the image, the missing factors can be filled in as follows:
- The green box should contain \((x - 5)\).
- The white box should contain \((x + 4)\).

Would you like further details or have any additional questions?

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Here are 5 related questions to consider:
1. How do you factor a quadratic expression like \(x^2 + 9x + 20\)?
2. Why can we cancel terms in rational expressions?
3. What are the general steps for factoring differences of squares?
4. How would you handle non-factorable quadratics?
5. Can all rational expressions be simplified?

**Tip**: Always check for common factors before multiplying expressions to simplify the process.

Use factoring to find the simplified product of these rational expressions: (x^2 - 49)/(x^2 + 9x + 20) * (x^2 - 25)/(x^2 - 6x - 7)

Learn how to factor and simplify the product of rational expressions like (x^2 - 49)/(x^2 + 9x + 20) * (x^2 - 25)/(x^2 - 6x - 7) by using factoring techniques such as difference of squares and factoring quadratics. Suitable for students in Grades 8-10.

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