Question 1030573
{{{(1+x^2)(2x+1/x)^10 }}}
={{{(1+x^2)sum(C(10,k)*(2x)^(10-k)*(1/x)^k, k = 0, 10)}}}

={{{sum(C(10,k)*(2x)^(10-k)*(1/x)^k, k = 0, 10) + x^2sum(C(10,k)*(2x)^(10-k)*(1/x)^k, k = 0, 10)}}}

= {{{sum(C(10,k)*2^(10-k)*x^(10-2k), k = 0, 10) + sum(C(10,k)*2^(10-k)*x^(12-2k), k = 0, 10)}}} 

In the first sum in the last expression, the term that is independent of x is when k = 5, i.e., {{{C(10, 5)*2^(10-5) = C(10, 5)*2^5}}}, while in the 2nd sum, is the term {{{C(10, 6)*2^(10-6) = C(10, 6)*2^4}}}.

Therefore, the term independent of x in the expansion of {{{(1+x^2)(2x+1/x)^10}}} is the constant term {{{ C(10, 5)*2^5 + C(10, 6)*2^4}}}.