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 To factor the expression \(x^4 + 4\), we can use a technique involving completing the square and then factoring by grouping. Here are the steps:

1. **Complete the square:**

   Notice that \(x^4 + 4\) can be written as:

   \[   x^4 + 4 = (x^2)^2 + 4x^2 + 4 - 4x^2   \]

   This is equivalent to:

   \[   (x^2)^2 + 4x^2 + 4 - 4x^2 = (x^2 + 2)^2 - (2x)^2   \]

2. **Factor using the difference of squares:**

   The expression \((x^2 + 2)^2 - (2x)^2\) is a difference of squares, which can be factored as:

   \[   (a^2 - b^2) = (a + b)(a - b)   \]

   Here, \(a = x^2 + 2\) and \(b = 2x\), so:

   \[   (x^2 + 2)^2 - (2x)^2 = (x^2 + 2 + 2x)(x^2 + 2 - 2x)   \]

3. **Simplify the factors:**

   The factored form of \(x^4 + 4\) is:

   \[   (x^2 + 2x + 2)(x^2 - 2x + 2)   \]

So, the factorization of \(x^4 + 4\) is:

\[(x^2 + 2x + 2)(x^2 - 2x + 2)\]

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