Quant Boosters  Hemant Malhotra  Set 11

general term = 8!/r1! * r2! * r3! * (1)^r1 * (x^2)^r2 * (x^3)^r3
= 8!/r1! * r2! * r3! * x^(2r1) * (1)^r3 * (x)^3r3
= 8!/r1! * r2! * r3! * (1)^r3 * x^(2r2+3r3)
now we have to find coefficient of x^6 here (for any coefficient same procedure )
so r1 + r2 + r3 = 8
2r2 + 3r3 = 6
find r1, r2 and r3 here
let r3=0 so r2=3 so r1=2 and let r3=2,so r2=0 so r1=6 (r1,r2,r3 should be integer because term can't be negative)
now coefficient will be 8!/r1! * r2! * r3! *(1)^r3
put both values of r1,r2 and r3 and add
so 8!/5! * 3!+8!/2! * 6!=84

@hemant_malhotra as the no. of real roots is asked so it will always be 3