To differentiate the function \( f(x) = (x^3 + 2x^2 + x)^4 \), we'll use the **chain rule**.

### Step 1: Identify the outer and inner functions
- The outer function is \( g(u) = u^4 \), where \( u = x^3 + 2x^2 + x \).
- The inner function is \( h(x) = x^3 + 2x^2 + x \).

### Step 2: Differentiate the outer function
- The derivative of the outer function \( g(u) = u^4 \) with respect to \( u \) is \( g'(u) = 4u^3 \).

### Step 3: Differentiate the inner function
- The derivative of the inner function \( h(x) = x^3 + 2x^2 + x \) with respect to \( x \) is:

\[
h'(x) = \frac{d}{dx} \left( x^3 + 2x^2 + x \right) = 3x^2 + 4x + 1
\]

### Step 4: Apply the chain rule
Using the chain rule:

\[
\frac{d}{dx} \left[ (x^3 + 2x^2 + x)^4 \right] = 4(x^3 + 2x^2 + x)^3 \cdot (3x^2 + 4x + 1)
\]

### Final Answer:
\[
\frac{d}{dx} \left[ (x^3 + 2x^2 + x)^4 \right] = 4(3x^2 + 4x + 1)(x^3 + 2x^2 + x)^3
\]

Do you need more details or have any questions?

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Here are some related questions for further exploration:

1. How would you differentiate \( (x^4 + x^3 + x^2 + x)^5 \)?
2. What is the derivative of \( (x^2 + x + 1)^6 \)?
3. Can you apply the chain rule to \( (2x^2 + 3x + 5)^3 \)?
4. What happens if you expand \( (x^3 + 2x^2 + x)^4 \) first and then differentiate?
5. How would you integrate \( 4(3x^2 + 4x + 1)(x^3 + 2x^2 + x)^3 \)?

**Tip:** When applying the chain rule, ensure you clearly identify both the outer and inner functions to correctly apply the derivative.

Learn how to differentiate (x^3 + 2x^2 + x)^4 using the chain rule with a detailed step-by-step solution. This problem involves applying the chain rule to a composite function and is suitable for advanced high school students and beyond.

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