<s>[INST] Problem of quadratic equation If $\alpha$ be a root of $ 4x^2 +2x -1 = 0 $ , prove that the other root is $4\alpha^3  - 3\alpha$ . I have tried to do it but of no success.[$4\alpha^3 -2\alpha$ = $\dfrac {-1}{2}$  and $4\alpha^4 - 3\alpha^2$ = $\dfrac {-1}{4}$ ] .How to prove it?
 [/INST] Write $4x^3-3x=(4x^2+2x-1)(x-1/2)-(x+1/2)$.
Plug $x=\alpha$ and get $4\alpha^3  - 3\alpha=-\alpha-1/2$, which is the other root because the sum of the roots is $-1/2$.
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