The simplification of $\sqrt{x^4 + 4x + 4}$ into $\sqrt{(x^2 + 2)(x^2 + 2)}$ involves identifying a perfect square inside the square root.

Here’s the step-by-step reasoning:

Step 1: Analyze the expression

We start with: $\sqrt{x^4 + 4x + 4}.$

The goal is to see if the term $x^4 + 4x + 4$ can be factored into a perfect square.

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Step 2: Factoring $x^4 + 4x + 4$

Looking at the structure of the expression $x^4 + 4x + 4$, notice it resembles a quadratic trinomial in $x^2$, which can be rewritten as: $(x^2)^2 + 4x^2 + 4.$

This simplifies further: $(x^2 + 2)^2.$

Thus: $x^4 + 4x + 4 = (x^2 + 2)^2.$

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Step 3: Substituting into the square root

Now substitute the factorized form back into the square root: $\sqrt{x^4 + 4x + 4} = \sqrt{(x^2 + 2)^2}.$

The square root of a square (under the assumption $x^2 + 2 \geq 0$) simplifies as: $\sqrt{(x^2 + 2)^2} = x^2 + 2.$

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Final Simplified Form

The simplification results in: $\sqrt{x^4 + 4x + 4} = x^2 + 2.$

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Note on the Simplification

• This works under the assumption that $x^2 + 2 \geq 0$, which is always true because $x^2 + 2 > 0$ for all real $x$.

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Related Questions:

1. What are the general rules for simplifying square roots of perfect squares?
2. How can you factorize expressions involving higher powers like $x^4$?
3. What are some methods to test if an expression is a perfect square?
4. Can $\sqrt{(x^2 + 2)^2}$ have negative outputs under certain conditions?
5. What is the geometric interpretation of squaring and square rooting functions?

Tip:

Always check if the terms inside a square root can be rewritten as a perfect square—it simplifies the evaluation significantly.