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To solve this problem, we need to find the coefficient of the second-to-last term in the binomial expansion of \( (x^2 + 2x)^8 \).

### Step 1: Identify the form of the expansion
We are expanding \( (x^2 + 2x)^8 \), which follows the binomial expansion formula:

\[
(a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k
\]

In this case:
- \( a = x^2 \)
- \( b = 2x \)
- \( n = 8 \)

The general term for the expansion is given by:

\[
T_k = \binom{8}{k} (x^2)^{8-k} (2x)^k
\]

### Step 2: Simplify the general term
Simplifying the term \( T_k \):

\[
T_k = \binom{8}{k} x^{2(8-k)} 2^k x^k = \binom{8}{k} 2^k x^{2(8-k) + k} = \binom{8}{k} 2^k x^{16 - k}
\]

### Step 3: Find the second-to-last term
To find the second-to-last term, we need to identify the power of \( x \) for that term. Since the highest power of \( x \) is 16, the second-to-last term will have \( x^2 \) as the exponent. Thus, we need \( 16 - k = 2 \), which gives \( k = 14 \).

However, this is impossible since \( k \) ranges from 0 to 8. Hence, the term with \( x^2 \) would be the third-to-last, and the second-to-last would be the term with \( x^4 \).

Thus, we find \( k \) where \( 16 - k = 4 \), giving \( k = 12 \).

This contradicts, so

Escribir el coeficiente del penúltimo término en el desarrollo de (x^2 + 2x)^8.

Learn how to find the coefficient of the second-to-last term in the binomial expansion of (x^2 + 2x)^8. This problem uses key concepts from the binomial theorem and simplifies the expansion for higher-degree polynomials. Suitable for students in Grades 10-12.

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