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The problem is asking you to multiply the binomial \( (3x^2 - 7)^2 \) using the rule for the square of a binomial.

The rule for the square of a binomial states:

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

Here, \( a = 3x^2 \) and \( b = 7 \). Now, apply the rule:

1. **Square the first term** \( a^2 \):
   \[
   (3x^2)^2 = 9x^4
   \]

2. **Multiply the first and second terms, then double it**:
   \[
   2 \times (3x^2) \times 7 = 42x^2
   \]

3. **Square the second term** \( b^2 \):
   \[
   7^2 = 49
   \]

So, putting it all together:

\[
(3x^2 - 7)^2 = 9x^4 - 42x^2 + 49
\]

This is the expanded form of the expression.

Would you like any further explanation or details?

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**Relative Questions:**
1. What is the square of a binomial formula?
2. How do you square a monomial like \(3x^2\)?
3. What happens if the signs in the binomial were different, like \( (3x^2 + 7)^2 \)?
4. Can you derive the formula for the square of a binomial?
5. What is the significance of the middle term in the expansion?
6. How can this formula be applied to more complex expressions?
7. What are the common mistakes made when expanding binomials?
8. How does this rule change for trinomials?

**Tip:** Always double-check each step in your binomial expansion to ensure accuracy, especially the sign of the middle term.

Learn how to expand the binomial (3x^2 - 7)^2 using the square of a binomial formula. This problem covers key algebraic concepts, including the binomial theorem and algebraic expansions. Suitable for high school students studying algebra.

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