Proving the second root of a quadratic equation

If $\alpha$ is a root of the equation $4x^2+2x-1=0$, then prove that $4\alpha^3-3\alpha$ is the other root. How do I proceed? The sum of the roots, the product of the roots lead me nowhere. Should I find the roots of the equation and substitute in the given two expressions of $\alpha$ and check whether manipulating the first root gives me the second root (which seems much complicated), or is there any other way?

Solution 1:

The other root $\beta$ is determined by the properties $\alpha+\beta=-\frac 12$ and $\alpha\beta=-\frac14$. Use polynomial division to show that $(4\alpha^3-3\alpha)+\alpha = [???]\cdot(4\alpha^2+2\alpha-1)-\frac12$ and $(4\alpha^3-3\alpha)\cdot\alpha = [???]\cdot(4\alpha^2+2\alpha-1)-\frac14$

Solution 2:

Set $x=4\alpha^2-3\alpha$ in the original polynomial equation, and check to see that the left side reduces to zero under the assumption that $4\alpha^2+2\alpha-1=0$, or equivalently that $\alpha^2 = \frac 14(-2\alpha+1)$.