To factorize the expression (x^2 + 2x) - (x^2 + 2x + 4), we need to simplify it step by step. Let's break it down into smaller parts and explain each step in detail.

Step 1: Distribute the negative sign
(x^2 + 2x) - (x^2 + 2x + 4) = x^2 + 2x - x^2 - 2x - 4

Step 2: Combine like terms
In the expression, we have x^2 terms, 2x terms, and a constant term (-4). Let's combine the like terms separately.

For the x^2 terms:
x^2 - x^2 = 0

For the 2x terms:
2x - 2x = 0

For the constant terms:
-4

So, the simplified expression becomes 0 - 0 - 4, which is equal to -4.

Therefore, the factorized form of (x^2 + 2x) - (x^2 + 2x + 4) is simply -4.

Let's summarize the steps:

1. Distribute the negative sign to all the terms inside the parentheses.
2. Combine like terms separately for x^2, 2x, and constant terms.
3. Simplify the expression by performing the arithmetic operations.
4. The final result is -4.

By following these steps, we have successfully factorized the given expression.