qwen2.5-coder

 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:

   $$\begin{array}{}\text{(1)}& x^4+4=(x^2{)}^2+4x^2+4-4x^2\end{array}$$

   This is equivalent to:

   $$\begin{array}{}\text{(2)}& (x^2{)}^2+4x^2+4-4x^2=(x^2+2{)}^2-(2x{)}^2\end{array}$$

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:

   $$\begin{array}{}\text{(3)}& (a^2-b^2)=(a+b)(a-b)\end{array}$$

   Here, $a=x^2+2$ and $b=2x$, so:

   $$\begin{array}{}\text{(4)}& (x^2+2{)}^2-(2x{)}^2=(x^2+2+2x)(x^2+2-2x)\end{array}$$

3. **Simplify the factors:**

   The factored form of $x^4+4$ is:

   $$\begin{array}{}\text{(5)}& (x^2+2x+2)(x^2-2x+2)\end{array}$$

So, the factorization of $x^4+4$ is:

$$\begin{array}{}\text{(6)}& (x^2+2x+2)(x^2-2x+2)\end{array}$$

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