Let's break down the expression and simplify the square root:

You are given:

$\sqrt{x^4 + 4x^2 + 4}$

Step 1: Recognize the form of the quadratic expression

Notice that the expression inside the square root resembles a perfect square trinomial. We can attempt to factor it as such.

The expression $x^4 + 4x^2 + 4$ looks similar to the expansion of $(x^2 + 2)^2$. Let's check if this is the case.

Step 2: Expand $(x^2 + 2)^2$

Expanding $(x^2 + 2)^2$ using the distributive property:

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

Step 3: Identify that it's a perfect square trinomial

From the expansion, we see that:

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

Step 4: Take the square root of both sides

Now that we know $x^4 + 4x^2 + 4 = (x^2 + 2)^2$, we can simplify the square root:

$\sqrt{x^4 + 4x^2 + 4} = \sqrt{(x^2 + 2)^2}$

Since the square root of a square returns the absolute value, we get:

$\sqrt{(x^2 + 2)^2} = |x^2 + 2|$

Final Answer:

Thus, the expression simplifies to:

$\sqrt{x^4 + 4x^2 + 4} = |x^2 + 2|$

Explanation of how it becomes $\sqrt{(x^2 + 2)(x^2 + 2)}$:

When you see $\sqrt{x^4 + 4x^2 + 4}$, it's common to factor it as $(x^2 + 2)^2$, which is then written as $\sqrt{(x^2 + 2)(x^2 + 2)}$. The two factors inside the square root are identical, thus giving the square of $x^2 + 2$.

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Would you like more details or a further explanation? Here are some related questions you might find useful:

1. How do you identify a perfect square trinomial?
2. What is the general method for simplifying square roots with binomial squares?
3. How can the absolute value affect the simplification of square roots?
4. What happens if we don't use absolute value when simplifying $\sqrt{(x^2 + 2)^2}$?
5. Can this process be applied to other polynomials?

Tip: Always check if the polynomial inside the square root can be factored as a perfect square trinomial, as it simplifies the process.