Let's factor the quadratic expression given for the area of the first rectangle, \( A_1 = x^2 + 6x - 13 \).

### Step 1: Factor the quadratic expression

The given expression \( A_1 = x^2 + 6x - 13 \) is a quadratic trinomial. To factor this, we need to look for two numbers that multiply to \( -13 \) (the constant term) and add to \( 6 \) (the coefficient of \( x \)).

However, \( x^2 + 6x - 13 \) does not factor nicely with integer coefficients. Therefore, instead of factoring, we can complete the square to simplify it.

### Step 2: Complete the square

To complete the square for \( x^2 + 6x - 13 \):

1. Start with the expression \( x^2 + 6x \).
2. Take half of the coefficient of \( x \), which is \( 6 \). Half of \( 6 \) is \( 3 \), and square it: \( 3^2 = 9 \).
3. Add and subtract \( 9 \) inside the expression:

\[
x^2 + 6x + 9 - 9 - 13
\]

4. Simplify the expression:

\[
(x + 3)^2 - 22
\]

So, the area \( A_1 = (x + 3)^2 - 22 \).

### Step 3: Interpretation

The expression \( A_1 = (x + 3)^2 - 22 \) represents the area of the first rectangle.

If we need to find the area of the second rectangle \( A_2 \), additional information about the relationship between \( A_1 \) and \( A_2 \) is required, such as the value of \( A_1 - A_2 \) or any specific properties of \( A_2 \). Without more details, we can't determine \( A_2 \) exactly, but the factored or completed square form of \( A_1 \) helps in simplifying or comparing it with any other expressions given.

Would you like to provide more details about \( A_2 \) or ask further questions based on this information? Here are some related queries that might help:

1. Could there be a geometric relationship between \( A_1 \) and \( A_2 \)?
2. Is \( A_2 \) possibly derived from \( A_1 \) by altering the width or length?
3. Are there any additional conditions or values given in the problem?
4. Could \( A_2 \) involve a known factor or constant in relation to \( A_1 \)?
5. Is there a specific method you are supposed to use to find \( A_2 \)?
6. Does \( A_2 \) require further algebraic manipulation?
7. Should we consider a comparison, like setting \( A_1 \) equal to \( A_2 \)?
8. Can the relationship between \( A_1 \) and \( A_2 \) involve solving an equation?

**Tip:** Completing the square is a powerful method to simplify quadratic expressions and find their minimum or maximum values, especially useful in geometry problems involving areas.

Learn how to proceed with factoring quadratic equations with a detailed step-by-step solution. This problem covers key algebraic concepts, including quadratic equations and completing the square. Suitable for students in Grades 9-12.

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