Question

How do you factor completely `2x^4 -14x^2 +24`?

Answer 1

`color(green)(2(x-sqrt(3))(x+sqrt(3))(x-2)(x+2))`

Explanation:

Temporally replacing `x^2` with `a`
`2x^4-14x^2+24` becomes `2a^2-14a+24`

`2a^2-14a+24`
`color(white)("XXX")=2(a^2-7a+12)`

`color(white)("XXX")=2(a-3)(a-4)`

Restoring `x^2` in place of `a`
#color(white)("XXX"){: ((a-3)=(x^2-3),color(white)("XX"),(a-4)=(x^2-4)), (color(white)("X")=(x-sqrt(3))(x+sqrt(3)),,color(white)("X")=(x-2)(x+2)) :}#

So
`2x^4-14x^2+24`
`color(white)("XXX")=color(green)(2(x-sqrt(3))(x+sqrt(3))(x-2)(x+2))`