understanding induction problems

Suppose that I am to prove by Inductive proof that F(n)G(n)=(2n)!, n≥1
and my F(n) is the (2k-1) from 1 to n and G(n) is 2k from 1 to n
How do i prove this? I try this so far by i got stuck please help.
P(n) = (2n-1)(2n)=(2n)! for n≥1
P(1) = (2*1-1)(2*1)=(2*1)!
(1)(2)=(2*1)!
2=2 TRUE

Assume P(n) Prove P(n+1)

Proof: P(n+1) = (2n-1)(2n)+(2(n+1)-1)*(2(n+1))= (2(n+1))!
= (2n)!+2n+2n+2 = (2n+2)!
=(2n)!+(4n+2)=(2n+2)!
I cannot get that left hand side to equal (2n+2)! anyway i tried.
What is the correct way to do this?
Also maybe i shouldn't replace F(n) by 2n-1 and G(n) by 2k by leave
them in as part of proof?

Thank you.
.