Factor the expression completely.x, to the power 4 , plus, 2, x, squared, minus, 8x 4 +2x 2 −8

The expression you provided is x^4 + 2x^2 - 8. To factor this completely, we can follow these steps:

1. First, recognize that this is a quadratic expression in the form of ax^2 + bx + c, where x^2 is replaced by y (let y = x^2). So, the expression becomes y^2 + 2y - 8.

2. Now, factor the quadratic expression. We are looking for two numbers that multiply to -8 (the c term) and add to 2 (the b term). The numbers 4 and -2 satisfy these conditions. So, we can write the expression as (y - 2)(y + 4).

3. Substitute y = x^2 back into the expression to get the completely factored expression: (x^2 - 2)(x^2 + 4).