https://r2.erweima.ai/mathbot/image/1726881174370zj0s5g5d.jpg

The problem extracted from the image is:

**"Al resolver el siguiente producto notable \( \left( 3x^2 + 4xy^7 \right)^2 \), se obtiene:"**

Options provided:

- a. \( 9x^4 + 24x^3y^7 + 16x^2y^{14} \)
- b. \( 9x^4 + 24x^2y^7 + 16x^2y^{14} \)
- c. \( 9x^4 + 16x^2y^{14} \)
- d. \( 9x^4 + 40x^5y^{21} \)

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To solve this, we need to expand the binomial:

\[
(3x^2 + 4xy^7)^2
\]

Using the formula for squaring a binomial:

\[
(a + b)^2 = a^2 + 2ab + b^2
\]

Let:
- \( a = 3x^2 \)
- \( b = 4xy^7 \)

Now, calculate each term:

1. \( a^2 = (3x^2)^2 = 9x^4 \)
2. \( b^2 = (4xy^7)^2 = 16x^2y^{14} \)
3. \( 2ab = 2(3x^2)(4xy^7) = 24x^3y^7 \)

Now, combining all terms:

\[
9x^4 + 24x^3y^7 + 16x^2y^{14}
\]

The correct answer is **option a**.

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Would you like more details on any step?

Here are some related questions:
1. What is the general rule for squaring a binomial?
2. How does the exponent on a variable affect the result in binomial expansion?
3. What are some other common notable products in algebra?
4. How does the distribution of multiplication work with terms involving exponents?
5. What is the relationship between terms with different exponents?

**Tip:** When expanding binomials, always apply the distributive property carefully to each pair of terms.

Al resolver el siguiente producto notable (3x^2 + 4xy^7)^2, se obtiene:

Learn how to solve the binomial expansion (3x^2 + 4xy^7)^2 with a detailed step-by-step solution. This algebraic problem covers key concepts in exponentiation, binomial expansion, and the application of the distributive property. Ideal for high school students studying algebra.

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