Final answer:
To factor x⁴ + 21x² - 100, substitute x² with a new variable y and then factor the resulting equation (y² + 21y - 100). Substitute x² back in for y to get the final factored form (x² + 25)(x² - 4).
Step-by-step explanation:
To factor the expression x⁴ + 21x² - 100, we can use a substitution to simplify the equation. Let's substitute x² with a new variable, let's say y. So the equation becomes y² + 21y - 100. We can now factor this equation by finding two numbers that add up to 21 and multiply to -100. The numbers are +25 and -4. Therefore, the factored form of the equation is (y + 25)(y - 4).
Now, let's substitute x² back in for y in the factored form. So we get (x² + 25)(x² - 4). This is the final factored form of the expression x⁴ + 21x² - 100.