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The given expression is:

\[
x^4 - 8x^2 + 15
\]

To factor this completely, let's treat this as a quadratic in terms of \(x^2\). We can rewrite the expression as:

\[
(x^2)^2 - 8(x^2) + 15
\]

Now, we will factor this like a quadratic equation. We need two numbers that multiply to give \(15\) and add up to \(-8\). These numbers are \(-3\) and \(-5\), so we can factor the expression as:

\[
(x^2 - 3)(x^2 - 5)
\]

This is the completely factored form of the given expression. It can't be factored further with real numbers, as both \(x^2 - 3\) and \(x^2 - 5\) are irreducible over the real numbers.

Would you like further details on how to approach similar problems?

Here are 5 related questions for practice:
1. How would you factor \(x^4 - 10x^2 + 25\)?
2. Can the expression \(x^2 - 2\) be factored over the real numbers?
3. What is the general method for factoring quadratic forms of higher-degree polynomials?
4. How do you solve equations that result from expressions like \(x^2 - 3 = 0\)?
5. How do complex numbers factor into expressions like \(x^4 + 4\)?

**Tip**: When factoring, always check if a higher-degree polynomial can be rewritten as a quadratic in terms of lower powers of \(x\), as this simplifies the factoring process!

Factor the expression completely: x^4 - 8x^2 + 15

Learn how to factor the expression x^4 - 8x^2 + 15 completely. This solution involves rewriting the polynomial as a quadratic in terms of x^2, then applying factoring techniques. Ideal for high school students studying Algebra or higher-level math.

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