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 treated 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 (the value of c) and add up to -5 (the value of b). The numbers that fit these conditions are -9 and 4.

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: Group the terms to factor by grouping: (x^4 - 9x^2) + (4x^2 - 36).

Step 5: Factor out the greatest common factor from each group: x^2(x^2 - 9) + 4(x^2 - 9).

Step 6: Notice that (x^2 - 9) is a common factor. Factor out (x^2 - 9) to get: (x^2 - 9)(x^2 + 4).

Step 7: The term (x^2 - 9) can be factored further as it is a difference of squares. This gives us: (x - 3)(x + 3)(x^2 + 4).

So, the expression x^4 - 5x^2 - 36 factors completely to (x - 3)(x + 3)(x^2 + 4).