Doubt from polynomial

@Sourav_Dey sir, @Samrat_2020

@Azimuddin_2019 sir please help..

Is this question from Pearson

Can you share your long method

@Sneha_2021 can you please tell what is the answer

24 and -3

Please share your method also

Let A=p^2 q+q^2r+r^2p and B=pq^2+qr^2+rp^2

We have \displaystyle A+B= \sum_{cyc} pq(p+q)= \sum_{cyc} pq(p+q+r-r)= \sum_{cyc} pq(6-r)

\displaystyle=6 \sum_{cyc} pq-3 pqr=21

\displaystyle A-B = \sum_{cyc} pq(p-q) = -(p-q)(q-r)(r-p)

\Rightarrow (A-B)^2=(p-q)^2(q-r)^2(r-p)^2 which is the discriminant of the given cubic which comes out to be 729. Hence A-B = \pm 27

Accordingly A=24 or A=-3


sir i m fan of u sir u have god level concept sir whenever u get time please share approches of ur i get lot of new concept even exploring in book to i dont find it.. sir thanks a lot for ur god level approches..

Thank you Sneha for the kind words