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The problem asks for the fourth term of the expansion of \( (3x^2 + x^3)^5 \), and it involves using the binomial theorem.

### Step 1: Use the Binomial Theorem
The binomial theorem is:
\[
(a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k
\]
In this case, \( a = 3x^2 \), \( b = x^3 \), and \( n = 5 \).

### Step 2: Identify the general term
The general term in the binomial expansion is given by:
\[
T_k = \binom{n}{k} (3x^2)^{n-k} (x^3)^k
\]

For this problem, \( n = 5 \), so the general term becomes:
\[
T_k = \binom{5}{k} (3x^2)^{5-k} (x^3)^k
\]

### Step 3: Find the fourth term
The fourth term corresponds to \( T_3 \), since the binomial expansion counts from \( T_0 \) (the first term). Let's compute \( T_3 \):

- \( \binom{5}{3} = \frac{5!}{3!(5-3)!} = 10 \)
- \( (3x^2)^{5-3} = (3x^2)^2 = 9x^4 \)
- \( (x^3)^3 = x^9 \)

So, the fourth term is:
\[
T_3 = 10 \cdot 9x^4 \cdot x^9 = 90x^{13}
\]

### Step 4: Check if this matches any options
The fourth term \( 90x^{13} \) does not match any of the given options. Therefore, the correct answer would be **e) Ninguno** (None).

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Would you like further details on this calculation or clarification on any step? 

Here are 5 related questions to expand on this problem:
1. How do you compute the binomial coefficient \( \binom{n}{k} \)?
2. What is the binomial expansion of \( (2x + 1)^4 \)?
3. How do you determine the degree of a term in a polynomial?
4. What are the key steps in applying the binomial theorem to polynomial expansions?
5. How does the expansion of \( (ax^m + bx^n)^p \) differ from standard binomial expansions?

**Tip**: Always identify the correct term (e.g., fourth term) by noting that binomial expansions start from \( T_0 \), not \( T_1 \).

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

Learn how to find the fourth term in the expansion of (3x^2 + x^3)^5 using the binomial theorem. This problem focuses on binomial expansion and polynomial manipulation, suitable for high school students in Grades 10-12.

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