Q.Solve $x^4 + 4x^3 - 2x^2 - 12x + 9 = 0$, given that it has two pairs of equal roots.

The roots are $1,1,-3,-3$.

If the roots are $\alpha,\alpha,\beta,\beta$, then

$x^4+4x^3-2x^2-12x+9=(x^2-sx+p)^2,$ where $s=\alpha+\beta,\ p=\alpha\beta.$

Expanding $(x^2-sx+p)^2=x^4-2sx^3+(s^2+2p)x^2-2spx+p^2$ and comparing:

$-2s=4\Rightarrow s=-2;\quad p^2=9\Rightarrow p=\pm3;\quad -2sp=-12\Rightarrow 4p=-12\Rightarrow p=-3.$

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