Question

Find the product:

`(x - 1/x + 4)(x - 1/x - 4)`

Answer 1

Given: `(x - 1/x + 4)(x - 1/x - 4)`

Step-wise calculation:

1. Recognize the expression as a product of the form (a + b)(a – b) = a² – b², where: 

`a = x - 1/x, b = 4`

2. Apply the identity:

`(x - 1/x + 4)(x - 1/x - 4) = (x - 1/x)^2 - 4^2`

`(x - 1/x + 4)(x - 1/x - 4) = (x - 1/x)^2 - 16`

3. Expand `(x - 1/x)^2`:

`(x - 1/x)^2 = x^2 - 2 xx x xx 1/x + 1/x^2`

`(x - 1/x)^2 = x^2 - 2 + 1/x^2`

4. Substitute back:

`(x - 1/x + 4)(x - 1/x - 4) = x^2 - 2 + 1/x^2 - 16`

`(x - 1/x + 4)(x - 1/x - 4) = x^2 + 1/x^2 - 18`