If `alpha` is a root of the equation `4x^2+2x-1=0,` then prove that `4alpha^3-3alpha` is the other root.

Explanation

Let other root is `beta`,
then `alpha+beta=-2/4=-1/2` or `beta=-1/2-alpha`……..i
and so `4alpha^(2)+2alpha-1=0` because `alpha` is a root of
`4x^(2)+2x-1=0`
Now `beta=4alpha^(3)-3alpha=alpha(4alpha^(2)-3)`
`=alpha(1-2alpha-3) [ :' 4 alpha^(2)+2alpha-1=0]`
`=-2 alpha^(2)-2 alpha`
`=-1/2(4alpha^(2))-2alpha`
`=-1/2(1-2alpha)-2alpha [ :' 4 alpha^(2)+2alpha-1=0]`
`=-1/2-alpha=beta` [ from eq. (i) ]
Hence `4alpha^(3)-3alpha` is the other root.