Factorize each of the following algebraic expressions:
(1) `a^4+3a^2+4`
(2) `4x^4+1`
(3) `4x^4+y^4`
(4) `(x+2)^2-6(x+2)+9`

Explanation

(1) `a^4+3a^2+4`
`=a^4+4a^2+4-a^2`
Using identity:`(a+b)^2=a^2+2ab+b^2`
`=[(a^2)^2+2timesa^2times2+(2)^2]-a^2`
`=[(a^2+2)^2-a^2`]
`=[(a^2+2)-a][(a^2+2)+a]`

(2) `4x^4+1`
`=4x^4+1+4x^2-4x^2`
Using identity:`(a+b)^2=a^2+2ab+b^2`
`=[(2x^2)^2+(1)^2+2times2x^2times1]-(2x)^2`
`=[(2x^2+1)^2-(2x)^2]`
`=(2x^2+1-2x)(2x^2+1+2x)`

(3) `4x^4+y^4`
`=4x^4+y^4+4x^2y^2-4x^2y^2`
`=[(2x^2)^2+(y^2)^2+2times2x^2timesy^2]-(2xy)^2`
`=[(2x^2+y^2)^2]-(2xy)^2`
`=[(2x^2+y^2-2xy)][(2x^2+y^2+2xy)]`

(4) `(x+2)^2-6(x+2)+9`
`=(x+2)^2-2times3times(x+2)+3`
`=(x+2-3)^2`
`=(x-1)^2`