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).

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 $T_1$.