Factor the expression completely.x, to the power 4 , minus, 10, x, squared, plus, 9x 4 −10x 2 +9

The expression you provided is x^4 - 10x^2 + 9. To factor this completely, we can treat it as a quadratic equation in the form of ax^2 + bx + c, where x^2 is treated as a single variable.

1. Identify a, b, and c in the equation x^4 - 10x^2 + 9 = 0. Here, a = 1, b = -10, and c = 9.

2. Factor the quadratic equation. We are looking for two numbers that multiply to 9 (ac) and add to -10 (b). The numbers -1 and -9 fit these criteria.

3. Rewrite the equation, replacing -10x^2 with -x^2 - 9x^2: x^4 - x^2 - 9x^2 + 9 = 0.

4. Factor by grouping: x^2(x^2 - 1) - 9(x^2 - 1) = 0.

5. Notice that (x^2 - 1) is a common factor: (x^2 - 1)(x^2 - 9) = 0.

6. Now, notice that both (x^2 - 1) and (x^2 - 9) are differences of squares, which can be factored further: (x - 1)(x + 1)(x - 3)(x + 3) = 0.

So, the expression x^4 - 10x^2 + 9 can be factored completely as (x - 1)(x + 1)(x - 3)(x + 3).