To simplify the given expression \( \sqrt{x^2+4x+4} \), we should first try to simplify the expression inside the square root.

1. Start by factoring the quadratic expression inside the square root:
   \[
   x^2 + 4x + 4
   \]
   Notice that this quadratic expression can be factored as:
   \[
   x^2 + 4x + 4 = (x + 2)^2
   \]

2. Substitute the factored form back into the square root:
   \[
   \sqrt{x^2 + 4x + 4} = \sqrt{(x + 2)^2}
   \]

3. Since the square root and the square are inverse operations, they cancel each other out. However, we must consider the absolute value to account for both positive and negative values:
   \[
   \sqrt{(x + 2)^2} = |x + 2|
   \]

Therefore, the simplified form of \( \sqrt{x^2 + 4x + 4} \) is \( |x + 2| \).

Would you like any more details or have any questions about this process? 

Here are some more questions you might find interesting:
1. How do you simplify \(\sqrt{x^2 + 6x + 9}\)?
2. Can you explain the properties of square roots and absolute values?
3. What are the steps to solve a quadratic equation by factoring?
4. How do you determine the vertex form of a quadratic equation?
5. What is the difference between solving an equation and simplifying an expression?

**Tip:** When simplifying square roots of quadratic expressions, always look to factor the quadratic expression first.

Learn how to simplify the square root of x^2 + 4x + 4 with a step-by-step solution. This problem covers key algebraic concepts, including square roots, quadratic equations, and absolute value theorem. Suitable for Grades 7-9.

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