https://file.aiquickdraw.com/mathbot/file/1727888718485mcqic6an.jpg

The integral shown in the image is:

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

### Solution:

1. **Step 1: Expand the expression inside the integral.**

We need to expand \((2x + 1)^2\):
\[
(2x + 1)^2 = (2x + 1)(2x + 1) = 4x^2 + 4x + 1
\]

Now, substitute this back into the integral:

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

2. **Step 2: Distribute the constant 2.**

\[
\int (8x^2 + 8x + 2) \, dx
\]

3. **Step 3: Integrate term by term.**

Now, integrate each term separately:

\[
\int 8x^2 \, dx = \frac{8x^3}{3}
\]
\[
\int 8x \, dx = 4x^2
\]
\[
\int 2 \, dx = 2x
\]

4. **Step 4: Combine the results.**

\[
\frac{8x^3}{3} + 4x^2 + 2x + C
\]

### Final Answer:

\[
\frac{8x^3}{3} + 4x^2 + 2x + C
\]

Would you like further details or clarification on any step?

### Relative Questions:
1. What is the method for expanding binomials like \((2x + 1)^2\)?
2. How do you perform integration on polynomial functions?
3. What is the general rule for integrating \(x^n\)?
4. How would you handle an integral involving a square root instead of a square?
5. How do you apply substitution in integrals like this?

**Tip:** Always simplify expressions before integrating to make the process easier.

Learn how to integrate the expression 2(2x + 1)^2 step by step. This solution covers polynomial expansion, integration of individual terms, and final result. Suitable for Grades 11-12 students studying calculus.

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