For a binomial expansion:
(x+y)^n we have:
((n),(r))x^(n-r)y^r
sum_(r=0)^(n)((n),(r))x^(n-r)y^r
Where:
((n),(r))=color(white)(0)^n C_(r)=(n!)/(r!(n-r)!)
Beginning with r=0
((5),(0))1^5(x^2)^0+((5),(1))1^4(x^2)^1+((5),(2))1^3(x^2)^2+((5),(3))1^2(x^2)^3
->((5),(4))1^1(x^2)^4+((5),(5))1^0(x^2)^5
Next calculate ((n),(r)) and simplify:
1+5x^2+10x^4+10x^6+5x^8+x^10
Short cut:
color(white)(0)^nC_(r)=color(white)(0)^nC_(n-r)
When one of the terms is 1, it is unnecessary to write out the powers. This was just done for completeness.