Factor the expression completely.x, to the power 4 , minus, 5, x, squared, minus, 36x 4 −5x 2 −36

The expression you provided is x^4 - 5x^2 - 36. To factor this completely, we can treat it as a quadratic equation in the form of ax^2 + bx + c, where x^2 is considered as a single variable.

Step 1: Identify a, b, and c in the equation. Here, a = 1 (coefficient of x^4), b = -5 (coefficient of x^2), and c = -36.

Step 2: Factor the quadratic equation. We are looking for two numbers that multiply to -36 (ac) and add to -5 (b). The numbers -9 and 4 fit these criteria because -9 * 4 = -36 and -9 + 4 = -5.

Step 3: Rewrite the middle term of the equation (b) as the sum of the products of a and each of the two numbers found in step 2. This gives us x^4 - 9x^2 + 4x^2 - 36.

Step 4: Factor by grouping. The first two terms can be grouped together, and the last two terms can be grouped together. This gives us x^2(x^2 - 9) + 4(x^2 - 9).

Step 5: Notice that (x^2 - 9) is a common factor. Factoring that out gives us (x^2 - 9)(x^2 + 4).

Step 6: The first term, x^2 - 9, is a difference of squares and can be factored further into (x - 3)(x + 3). The second term, x^2 + 4, cannot be factored further because it is a sum of squares.

So, the completely factored expression is (x - 3)(x + 3)(x^2 + 4).