# Can not 'see' how to get next line of a particular Sturm Sequence

Stu(0)P = X^4 + pX^2 + qx + r

Stu(1)P = 4x^3 + 2px + q

Stu(2)P = -[2px^2 + 3qx + 4r]/4

Should anyone know how to get from the 3 lines above to Stu(3)P shown here on next line:-

Stu^3(P) = -[(2p^3 - 8pr + 9q^2)x + (p^2)q + 12qr)]/(p^2)

I've read Wikipedia and the relevant webpages emphasising the importance of the minus sign in the remainder, and found the general notation of how one line leads on to the next with appropriate quotients factored in etc

Can anyone please show me the answer by demonstration no matter how simple/obvious/cumbersome the process?

Above taken from here.

Thanks

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Well, polynomial long division of $\mathrm{Stu}^1(P)$ by $\mathrm{Stu}^2(P)$ gives $$4x^3 + 2px + q = \left(-\frac{2px^2 + 3qx + 4r}{4}\right) \left( \frac{-8px + 12 q}{p^2} \right) + \frac{(2p^3 - 8pr + 9q^2)x + p^2 q + 12qr}{p^2}.$$ The last term is the remainder, $-\mathrm{Stu}^3(P)$.