1）4a^2+4a-4a^2b+b+1

1）4a^2+4a-4a^2b+b+1
＝(4a^2+4a+1)-(4a^2b-b)
=(2a+1)^2-b(2a+1)(2a-1)
=(2a+1)〔2a+1-b(2a-1)〕
=(2a+1)(2a+1-2ab+b)
2）ax^2+16ay^2-a-8axy
=a(x^2+16y^2-1-8xy)
=a〔(x^2+16y^2-8xy)-1〕
=a〔(x-4y)^2-1^2〕
=a(x-4y+1)(x-4y-1)
(3) a(a^2-a-1)+1
=a^3-a^2-a+1
=(a^3-a^2)-(a-1)
=a^2(a-1)-(a-1)
=(a^2-1)(a-1)
=(a+1)(a-1)^2
(4)ab(m^2+n^2)+mn(a^2+b^2)
=abm^2+abn^2+mna^2+mnb^2
=(abm^2+mna^2)+(mnb^2+abn^2)
=am(bm+an)+bn(bm+an)
=(am+bn)(bm+an)