To solve the expression \((2x + 1)^4\), we will expand it using the Binomial Theorem.

### Step 1: Understand the Binomial Theorem
The Binomial Theorem states that for any positive integer \( n \):

\[
(a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k
\]

In this case:
- \(a = 2x\)
- \(b = 1\)
- \(n = 4\)

### Step 2: Apply the Binomial Theorem
Expanding \((2x + 1)^4\):

\[
(2x + 1)^4 = \sum_{k=0}^{4} \binom{4}{k} (2x)^{4-k} (1)^k
\]

Let's calculate each term in the expansion:

- For \(k = 0\):
  \[
  \binom{4}{0} (2x)^4 (1)^0 = 1 \cdot 16x^4 \cdot 1 = 16x^4
  \]

- For \(k = 1\):
  \[
  \binom{4}{1} (2x)^3 (1)^1 = 4 \cdot 8x^3 \cdot 1 = 32x^3
  \]

- For \(k = 2\):
  \[
  \binom{4}{2} (2x)^2 (1)^2 = 6 \cdot 4x^2 \cdot 1 = 24x^2
  \]

- For \(k = 3\):
  \[
  \binom{4}{3} (2x)^1 (1)^3 = 4 \cdot 2x \cdot 1 = 8x
  \]

- For \(k = 4\):
  \[
  \binom{4}{4} (2x)^0 (1)^4 = 1 \cdot 1 \cdot 1 = 1
  \]

### Step 3: Combine all the terms
The expansion is:

\[
(2x + 1)^4 = 16x^4 + 32x^3 + 24x^2 + 8x + 1
\]

### Conclusion
So, the expanded form of \((2x + 1)^4\) is:

\[
\boxed{16x^4 + 32x^3 + 24x^2 + 8x + 1}
\]

Would you like more details on the binomial expansion? Here are some related questions you can try:

1. How would you expand \((3x - 2)^3\) using the Binomial Theorem?
2. What is the general term in the expansion of \((a + b)^n\)?
3. Can you expand \((x + 1/x)^4\) using the same method?
4. How do the coefficients in the binomial expansion relate to Pascal's Triangle?
5. What happens when \(x\) is a complex number in the binomial expansion?

**Tip:** When expanding binomials, make sure to correctly calculate each binomial coefficient using \(\binom{n}{k}\) and carefully apply the powers to each term.

Learn how to expand (2x + 1)^4 using the Binomial Theorem with a step-by-step detailed solution. This problem covers the application of the Binomial Theorem in algebra, providing clarity on binomial expansions. Suitable for advanced high school students and college-level algebra enthusiasts.

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