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Let's solve the questions step by step.

### Question 1: 
You are asked to find the product of \((n + 13)^2\).

This expression is a perfect square trinomial, and it can be expanded using the formula:
\[
(a + b)^2 = a^2 + 2ab + b^2
\]
For \((n + 13)^2\):
- \(a = n\)
- \(b = 13\)

Applying the formula:
\[
(n + 13)^2 = n^2 + 2(n)(13) + 13^2
\]
\[
= n^2 + 26n + 169
\]

So, the expanded form is:
\[
n^2 + 26n + 169
\]

Thus, the correct answer is the third option: **\(n^2 + 26n + 169\)**.

### Question 2:
You are asked to find the product of \((2k + 1)^3\).

This is a cube expression and can be expanded using the formula:
\[
(a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3
\]
For \((2k + 1)^3\):
- \(a = 2k\)
- \(b = 1\)

Applying the formula:
\[
(2k + 1)^3 = (2k)^3 + 3(2k)^2(1) + 3(2k)(1^2) + 1^3
\]
\[
= 8k^3 + 3(4k^2) + 3(2k) + 1
\]
\[
= 8k^3 + 12k^2 + 6k + 1
\]

So, the expanded form is:
\[
8k^3 + 12k^2 + 6k + 1
\]

If this option is available in the quiz, that is the correct answer.

Would you like further details on any step or have any questions about this? Here are five questions to deepen your understanding:

1. How would the expansion change for \((n - 13)^2\)?
2. What happens if we expand \((2k - 1)^3\) instead?
3. Can you generalize the expansion of \((a + b)^2\) to higher powers?
4. How does the binomial theorem apply to \((2k + 1)^3\)?
5. What are some real-world applications of expanding binomials?

**Tip:** Always verify your answers by plugging in small values for \(n\) or \(k\) and checking if the results match.

Find the product of (n + 13)^2 and (2k + 1)^3.

Learn how to expand and find the product of the expressions (n + 13)^2 and (2k + 1)^3 using polynomial expansion and the binomial theorem. This problem covers key algebraic concepts and is suitable for students in grades 9-11.

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