To expand the binomial \((x + 2)^{10}\), we use the Binomial Theorem. The Binomial Theorem states:

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

For \((x + 2)^{10}\):

- \( a = x \)
- \( b = 2 \)
- \( n = 10 \)

Thus, the expansion is:

\[
(x + 2)^{10} = \sum_{k=0}^{10} \binom{10}{k} \cdot x^{10-k} \cdot 2^k
\]

I'll calculate the expansion step by step.The expanded form of \((x + 2)^{10}\) is:

\[
x^{10} + 20x^9 + 180x^8 + 960x^7 + 3360x^6 + 8064x^5 + 13440x^4 + 15360x^3 + 11520x^2 + 5120x + 1024
\]

Would you like a detailed explanation of this expansion or have any other questions? Here are five related questions you might find useful:

1. How can you apply the Binomial Theorem to negative exponents?
2. What are the coefficients in the expansion of \((x - 3)^8\)?
3. How is the Binomial Theorem related to Pascal's Triangle?
4. Can you find the middle term in the expansion of \((x + 1)^7\)?
5. How would the expansion change if the binomial were \((2x + 3)^{10}\)?

**Tip:** The coefficients of each term in a binomial expansion correspond to the entries in Pascal's Triangle for that power.

Learn how to expand the binomial (x + 2)^10 using the Binomial Theorem with a detailed step-by-step solution. This problem covers key concepts in binomial expansion and the application of the Binomial Theorem. Suitable for advanced high school students.

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