To factor the expression x^4 + x^2 - 12, we can start by looking for common factors among the terms. In this case, we don't have any common factors like numbers or variables.
To proceed, we can try factoring the expression by grouping. Let's group the terms together:
(x^4 + x^2) - 12
Now, let's try to factor out a common factor from each group separately.
In the first group, we can factor out x^2, giving us:
x^2(x^2 + 1) - 12
Now, we have a difference of squares in the first group, which can be factored further. The difference of squares formula states that:
a^2 - b^2 = (a + b)(a - b)
In our case, a = x and b = 1, so we have:
x^2(x + 1)(x - 1) - 12
Simplifying further, we have the fully factored expression:
(x + 1)(x - 1)(x^2 - 12)
So, the correct factorization of x^4 + x^2 - 12 is (x + 1)(x - 1)(x^2 - 12).