Hello Sir,

I haven't solved the problem

but I think this can help a bit for the second part

a+b+c=6

b+bc+ca=9

(a+b+c)^2 = 36

a^2+b^2+c^2 +2(ab+bc+ca)=36

a^2+b^2+c^2=18

now

a^3+b^3+c^3 =(a+b+c)(a^2+b^2+c^2-ab-bc-ca) +3abc

=6(18-9)+3abc

=54+3abc

I stopped here

But I think

using a cubic equation like

x^3+6x^2+9x+m=0

can help

where m=abc