Can we show $ x^4 - px^3 - 6x^2 + px +1 =0$ always has four real roots?

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If one root of the equation \[ x^4 - px^3 - 6x^2 + px +1 =0\] is \(\alpha\), prove that the others are \[-\frac{1}{\alpha},\frac{\alpha-1}{\alpha+1},\frac{1+\alpha}{1-\alpha}.\]

Hence or otherwise show that for all real values of \(p\) the equation has four real roots.