## Understanding the Binomial Expansion of (x+2)^2

The binomial expansion is a powerful tool in algebra that allows us to expand expressions of the form (a + b)^n. In this case, we'll focus on expanding **(x + 2)^2**.

### Applying the Binomial Theorem

The binomial theorem states:

**(a + b)^n = a^n + (n choose 1)a^(n-1)b + (n choose 2)a^(n-2)b^2 + ... + (n choose n-1)ab^(n-1) + b^n**

where (n choose k) represents the binomial coefficient, calculated as:

**(n choose k) = n! / (k! * (n-k)!)**

Let's apply this to our expression, (x + 2)^2:

**a = x****b = 2****n = 2**

Now, let's break down the expansion step-by-step:

**(n choose 0)a^n b^0 = (2 choose 0)x^2 * 2^0 = 1 * x^2 * 1 = x^2****(n choose 1)a^(n-1)b^1 = (2 choose 1)x^1 * 2^1 = 2 * x * 2 = 4x****(n choose 2)a^(n-2)b^2 = (2 choose 2)x^0 * 2^2 = 1 * 1 * 4 = 4**

Therefore, the complete binomial expansion of (x + 2)^2 is:

**(x + 2)^2 = x^2 + 4x + 4**

### Alternative Approach

Instead of using the binomial theorem, you can also expand (x + 2)^2 by directly applying the distributive property:

(x + 2)^2 = (x + 2)(x + 2)

Expanding the product:

x(x + 2) + 2(x + 2) = x^2 + 2x + 2x + 4 = x^2 + 4x + 4

This method confirms the same result obtained using the binomial theorem.

### Conclusion

Understanding the binomial expansion is crucial for simplifying complex algebraic expressions. By applying the binomial theorem or using the distributive property, we can efficiently expand expressions like (x + 2)^2 and obtain the correct result.