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The problem asks: "¿Con qué expresiones podría construirlo? ¿Por qué?" (Which expressions could Geronimo use to construct it, and why?)

The key here is that Geronimo wants to build a square. The expression that would represent the area of a square must be a perfect square trinomial.

Let's analyze the given expressions to identify any perfect square trinomials:

1. **a. \( 9w^2 + 6wz + z^2 \)**  
   This can be rewritten as \( (3w + z)^2 \), which is a perfect square trinomial.

2. **b. \( 25x^2 - 60xy + 36y^2 \)**  
   This is also a perfect square trinomial, as it can be rewritten as \( (5x - 6y)^2 \).

3. **c. \( 25x^2 - 35xy + 49y^2 \)**  
   This is not a perfect square trinomial. The middle term does not match the pattern required for a perfect square trinomial.

4. **d. \( 9x^2 - 12yx + 4y^2 \)**  
   This is a perfect square trinomial because it can be rewritten as \( (3x - 2y)^2 \).

### Conclusion:
Geronimo could use expressions **a**, **b**, and **d** to construct the square because they represent perfect square trinomials. Expression **c** does not form a perfect square, so it cannot be used.

Would you like to know more details about perfect square trinomials or how to recognize them?

Here are five follow-up questions to further expand your understanding:
1. What makes a trinomial a perfect square trinomial?
2. How can you factor other types of quadratic trinomials?
3. Why is the middle term critical in recognizing a perfect square trinomial?
4. Can a square have more than one possible algebraic expression for its area?
5. What are some common mistakes when identifying perfect square trinomials?

**Tip:** When looking for perfect square trinomials, check if the first and last terms are squares and if the middle term is twice the product of their square roots!

Gerónimo desea construir un cuadrado y tiene las siguientes expresiones: 
a. 9w^2 + 6wz + z^2
b. 25x^2 - 60xy + 36y^2
c. 25x^2 - 35xy + 49y^2
d. 9x^2 - 12yx + 4y^2
¿Con qué expresiones podría construirlo? ¿Por qué?

Discover how to identify perfect square trinomials and determine which expressions can be used to construct a square. This problem involves factoring trinomials and understanding algebraic concepts. Suitable for Grades 8-10 students.

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