Simplifying and Factoring (x+2)^2  4
This expression, (x+2)^2  4, can be simplified and factored in a couple of different ways. Let's explore both methods:
Method 1: Expanding and Simplifying

Expand the square: (x+2)^2 is the same as (x+2)(x+2). Using the FOIL method (First, Outer, Inner, Last), we get: (x+2)(x+2) = x^2 + 2x + 2x + 4 = x^2 + 4x + 4

Combine with the constant term: Now we have x^2 + 4x + 4  4

Simplify: The +4 and 4 cancel out, leaving us with x^2 + 4x.
Method 2: Factoring by Difference of Squares

Recognize the pattern: The expression is in the form of a^2  b^2, where a = (x+2) and b = 2.

Apply the difference of squares formula: This formula states that a^2  b^2 = (a + b)(a  b).

Substitute and simplify: Applying this to our expression, we get: (x+2)^2  4 = (x+2 + 2)(x+2  2) = (x+4)(x)
Conclusion
Both methods lead to the same simplified and factored form of the expression: x^2 + 4x or (x+4)(x).
The method you choose depends on your preference and the specific problem you are trying to solve. Sometimes, expanding and simplifying is more convenient, while other times, factoring by difference of squares offers a more direct approach.