We have:
#P(x) = sum_(n=0)^6 c_nx^n = (1+3x)^6(1-3x-5x^2)#
Develop #(1+3x)^6# using binomial coefficients:
#sum_(n=0)^6 c_nx^n = (1-3x-5x^2)sum_(n=0)^6 ((6),(n))(3x)^n#
Now using the distributive property of multiplication:
#sum_(n=0)^6 c_nx^n = sum_(n=0)^6 ((6),(n))3^nx^n + sum_(n=0)^6 ((6),(n))(-3x)3^nx^n +sum_(n=0)^6 ((6),(n))(-5x^2)3^nx^n#
#sum_(n=0)^6 c_nx^n = sum_(n=0)^6 ((6),(n))3^nx^n - sum_(n=0)^6 ((6),(n))3^(n+1)x^(n+1) -5sum_(n=0)^6 ((6),(n))3^nx^(n+2)#
The coefficient of #x^2# is then the sum of the term for #n=2# in the first sum, for #n=1# in the second sum and for #n=0# in the third sum:
#c_2 = ((6),(2)) * 3^2 - ((6),(1)) * 3^2 -5*((6),(0))#
The general expression of the binomial coefficient is:
#((n),(k)) = (n!)/(k!(n-k)!)#
So:
#((6),(2)) = (6!)/((2!)(4!)) = 15#
#((6),(1)) = (6!)/((1!)(5!)) = 6#
#((6),(0)) = (6!)/((0!)(6!)) = 1#
and:
#c_2 = 15 * 9 - 6 * 9 -5 = 76#
