To factor the expression x^4 + 8x^2 + 12, we can first notice that it is a quadratic in terms of x^2. We can rewrite it as (x^2)^2 + 8(x^2) + 12.

Now, let's try to factor this quadratic expression. We can use the fact that the sum of the roots of a quadratic equation ax^2 + bx + c = 0 is -b/a and the product of the roots is c/a.

In this case, the sum of the roots is -8 and the product of the roots is 12. We need to find two numbers that add up to -8 and multiply to 12.

The numbers that satisfy these conditions are -6 and -2. Therefore, we can rewrite the expression as:

(x^2 - 6)(x^2 - 2).

So, the factored form of x^4 + 8x^2 + 12 is (x^2 - 6)(x^2 - 2).