Final Answer:
The real roots of the function f(x) = -(x² - 1)(x² + 2) are x = -1, x = 1.
Step-by-step explanation:
The given function f(x) = -(x² - 1)(x² + 2) can be factored into two quadratic expressions: -(x² - 1) and (x² + 2). The roots of the first quadratic -(x² - 1) can be found by setting x² - 1 equal to zero and solving for x:
x² - 1 = 0
Adding 1 to both sides gives x² = 1, and taking the square root of both sides yields x = ±1. Therefore, the roots of -(x² - 1) are x = -1 and x = 1.
Similarly, for the second quadratic (x² + 2), setting x² + 2 equal to zero and solving for x gives:
x² + 2 = 0
Subtracting 2 from both sides gives x² = -2, but since the square of any real number is non-negative, this quadratic has no real roots. Therefore, the only real roots of the entire function f(x) come from the first quadratic, resulting in x = -1 and x = 1.
In conclusion, the real roots of f(x) = -(x² - 1)(x² + 2) are x = -1 and x = 1.