Factorize each of the following algebraic expressions:
(1) `9a^4-24 a^2b^2+16 b^4-256`
(2)`16-a^6+4a^3b^3-4b^6`
(3)`a^2-2a b+b^2-c^2`
(4) `x^2+2x+1-9y^2`

Explanation

(1) `9a^4-24a^2b^2+16b^4-256`
Using identity:`(a-b)^2=a^2-2ab+b^2`
`=(9a^4-24a^2b^2+16b^4)-256`
`=[(3a^2)^2-2times3a^2times4b^2+(4b^2)^2]-16^2`
`=(3a^2-4b^2)^2-16^2`
`=[(3a^2-4b^2)-16][(3a^2-4b^2)+16]`

(2) `16-a^6+4a^3b^3-4b^6`
`=16-(a^6+4a^3b^3)-4b^6`
`=4^2-[(a^3)^2-2timesa^3times2b^3+(2b^3)^2]`
`=[4^2-(a^3-2b^3)^2]`
`=(4-a^3+2b^3)(4+a^3-2b^3)`

(3) `a^2-2a b+b^2-c^2`
Using identity:`(a-b)^2=a^2-2ab+b^2`
`=(a^2-2ab+b^2)-c^2`
`=[(a^2-2timesatimesb+b^2)]-c^2`
`=(a-b)^2-c^2`
`=[(a-b)-c][(a-b)+c]`

(4) `x^2+2x+1-9y^2`
Using identity:`(a+b)^2=a^2+2ab+b^2`
`=(x^2+2x+1^2)-9y^2`
`=[(x^2+2timesxtimes1+1^2)]-(3y)^2`
`=(x+1)^2-(3y)^2`
`=[(x+1)-3y][(x+1)+3y]`