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The problem asks us to expand \((3x^2 + x^3)^5\) and find the fourth term of the resulting polynomial, ordered in descending powers of \(x\).

### Step-by-step approach:

We will use the **binomial theorem**, which states:
\[
(a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k
\]
Here, \(a = 3x^2\), \(b = x^3\), and \(n = 5\).

So, the general term in the expansion is:
\[
\binom{5}{k} (3x^2)^{5-k} (x^3)^k
\]
Let's compute each part:

1. **Binomial coefficient**: \(\binom{5}{k}\)
2. **First term**: \((3x^2)^{5-k} = 3^{5-k} x^{2(5-k)}\)
3. **Second term**: \((x^3)^k = x^{3k}\)

Thus, the general term becomes:
\[
\binom{5}{k} \cdot 3^{5-k} \cdot x^{2(5-k)} \cdot x^{3k} = \binom{5}{k} \cdot 3^{5-k} \cdot x^{10-2k+3k} = \binom{5}{k} \cdot 3^{5-k} \cdot x^{10+k}
\]

Now let's calculate the first few terms of the expansion:

- **For \(k = 0\)**:
  \[
  \binom{5}{0} \cdot 3^5 \cdot x^{10+0} = 1 \cdot 243 \cdot x^{10} = 243x^{10}
  \]

- **For \(k = 1\)**:
  \[
  \binom{5}{1} \cdot 3^4 \cdot x^{11} = 5 \cdot 81 \cdot x^{11} = 405x^{11}
  \]

- **For \(k = 2\)**:
  \[
  \binom{5}{2} \cdot 3^3 \cdot x^{12} = 10 \cdot 27 \cdot x^{12} = 270x^{12}
  \]

- **For \(k = 3\)**:
  \[
  \binom{5}{3} \cdot 3^2 \cdot x^{13} = 10 \cdot 9 \cdot x^{13} = 90x^{13}
  \]

The terms so far are:
\[
243x^{10}, 405x^{11}, 270x^{12}, 90x^{13}, \dots
\]

Since the problem asks for the **fourth term**, and we order the polynomial in **decreasing powers of \(x\)**, the fourth term is \(405x^{11}\).

### Answer:
The correct option is **b) 405 \(x^{11}\)**.

Let me know if you need further details or explanations.

Here are 5 related questions to further your understanding:
1. How does the binomial theorem apply to polynomial expansions?
2. How do you calculate binomial coefficients for any given \(n\) and \(k\)?
3. What is the significance of ordering terms in descending powers of \(x\)?
4. Can you use the binomial theorem to expand \((2x + 5)^4\)?
5. How do powers of terms change in binomial expansions involving polynomials?

**Tip**: When expanding using the binomial theorem, remember to simplify both the coefficients and the powers of each term for easier calculation.

Luego de desarrollar: (3x^2 + x^3)^5 y ordenar el polinomio resultante de forma decreciente, el cuarto término del desarrollo es:

Find the fourth term of the polynomial expansion of (3x^2 + x^3)^5 using the binomial theorem. Detailed step-by-step solution covers algebraic expansion and simplification techniques. Suitable for high school students.

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