If the polynomial 4A ^ 2 + 1 plus a monomial form becomes a complete square form, what can the monomial form be?

It can be directly decomposed into (2a + 1) (2a-1)
perhaps
4a2+1=4(a2+ 1/4)=4(a2+a+ 1/4 -a)=4[(a+ 1/2 )2-a]=4(a+ 1/2 )2-4a

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