Question

How do you factor `y= x^4 + 5x^2 - 6` ?

Answer 1

`y=(x^2+6)(x+1)(x-1)`

Explanation:

Given `y=x^4+5x^2-6`

We notice that all terms containing the variable `x` have `x^2` as a factor,
so to simplify things initially we will replace `x^2` with `z`

`color(white)("XXX")y=z^2+5z-6`
which can easily be factored as:
`color(white)("XXX")y=(z+6)(z-1)`

Restoring `x^2` back in place of `z`
`color(white)("XXX")y=(x^2+6)(x^2-1)`

The first factor, `(x^2+6)`, has no obvious sub-factors
but the second, `(x^2-1)`, is the difference of squares with sub-factors `(x+1)(x-1)`

giving
`color(white)("XXX")y=(x^2+6)(x+1)(x-1)`

Answer 2

`y = (x^2 + 6)(x+1)(x-1)`

Explanation:

Perhaps it will help if I rewrite this expression for `y` in a slightly different way:

`y = (x^2)^2 + 5(x^2) - 6`

Let `u = x^2`. Then we have

`y = u^2 + 5u - 6`

This is something we can factor pretty easily.

`y = (u + 6)(u - 1)`

Just substitute back to get `y` in terms of `x`:

`y = (x^2 + 6)(x^2 - 1)`

And now, the last step is to factor the second bit which is a difference of two squares:

`y = (x^2 + 6)(x - 1)(x+1)`