Question

What is the vertex form of `y=x^4+2x^2-2`?

Answer 1

I think the best we can do is
`color(white)("XXX")y=(x^2+1)^2-3`

Explanation:

A reference to "the vertex form" implies that we are dealing with a parabolic equation; `y=x^4+2x^2-2` is not a parabolic equation.

We could replace `x^2` with `z`, so
`color(white)("XXX")y=z^2+2z-2`

`color(white)("XXX")y=(z^2+2zcolor(red)(+1)-2color(red)(-1)`
`color(white)("XXX")y=(z+1)^2-3`
which is a parabolic equation with the vertex at `(color(blue)(z),y) = (-1,-3)`

Replacing `z` with `x^2` gives
`color(white)("XXX")y=(x^2+1)-3`
but as noted, this is not a parabola (see graph below)
graph{x^4-2x^2-2 [-6.783, 7.26, -4.515, 2.505]}