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The given integral is:

\[
\int (2x + 1)^2 \, dx
\]

### Step 1: Expand the integrand
First, expand \((2x + 1)^2\):

\[
(2x + 1)^2 = (2x + 1)(2x + 1) = 4x^2 + 4x + 1
\]

### Step 2: Integrate term by term
Now, we integrate each term of the expanded expression:

\[
\int (4x^2 + 4x + 1) \, dx = \int 4x^2 \, dx + \int 4x \, dx + \int 1 \, dx
\]

1. \(\int 4x^2 \, dx = \frac{4}{3}x^3\)
2. \(\int 4x \, dx = 2x^2\)
3. \(\int 1 \, dx = x\)

### Step 3: Combine the results
Now, combine all the terms:

\[
\frac{4}{3}x^3 + 2x^2 + x + C
\]

Thus, the solution to the integral is:

\[
\int (2x + 1)^2 \, dx = \frac{4}{3}x^3 + 2x^2 + x + C
\]

Would you like further clarification on any of the steps? 

Here are some related questions for deeper exploration:

1. How would the solution change if the exponent were 3 instead of 2?
2. What is the general method for solving integrals of polynomials?
3. How can integration by parts be applied to more complex expressions?
4. How does the Fundamental Theorem of Calculus connect to this integral?
5. Can this integral be solved using a substitution method? How?

**Tip:** Always expand polynomial expressions before integrating them term by term for simpler results.

Explore the detailed steps to solve the integral ∫(2x + 1)^2 dx. This problem involves expanding polynomials and applying basic integration rules. Suitable for students in Grades 10-12.

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