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The given integral is:

\[
\int_{0}^{1} 4x(1 - x^2) \cdot x^2 \, dx
\]

### Step 1: Simplify the Integrand

First, let's simplify the expression inside the integral:

\[
4x(1 - x^2) \cdot x^2 = 4x \cdot x^2 \cdot (1 - x^2)
\]

Expanding this:

\[
4x \cdot x^2 \cdot (1 - x^2) = 4x^3(1 - x^2)
\]

Now, distribute the \(4x^3\):

\[
4x^3 - 4x^5
\]

### Step 2: Set up the Integral

The integral now becomes:

\[
\int_{0}^{1} (4x^3 - 4x^5) \, dx
\]

### Step 3: Integrate

Now, find the antiderivative of each term:

1. The antiderivative of \(4x^3\) is \(x^4\).
2. The antiderivative of \(-4x^5\) is \(-\frac{2}{3}x^6\).

So, the antiderivative of the entire expression is:

\[
x^4 - \frac{2}{3}x^6
\]

### Step 4: Evaluate the Definite Integral

Now, evaluate this expression from 0 to 1:

\[
\left[ x^4 - \frac{2}{3}x^6 \right]_{0}^{1}
\]

Substituting the upper limit (1):

\[
1^4 - \frac{2}{3} \cdot 1^6 = 1 - \frac{2}{3} = \frac{1}{3}
\]

Substituting the lower limit (0):

\[
0^4 - \frac{2}{3} \cdot 0^6 = 0
\]

### Step 5: Final Answer

Subtract the lower limit from the upper limit:

\[
\frac{1}{3} - 0 = \frac{1}{3}
\]

Thus, the value of the integral is:

\[
\boxed{\frac{1}{3}}
\]

---

Would you like further details or explanations on any step? 

Here are five related questions:
1. How do you find the antiderivative of more complex expressions?
2. What is the geometric interpretation of definite integrals?
3. How can you use substitution in integrals?
4. What happens when you change the limits of integration?
5. How do you evaluate improper integrals?

**Tip:** Always double-check your antiderivatives by differentiating them to ensure accuracy.

Evaluate the definite integral from 0 to 1 of 4x(1 - x^2) * x^2 dx.

This solution walks through how to evaluate the definite integral from 0 to 1 of the function 4x(1 - x^2) * x^2 dx. Using polynomial expansion and basic integration formulas, the integral is simplified and evaluated step by step.

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