Question

Please help with this question on factorisation. Please show steps because I want to understand.

Answer 1

`(x - 2y)((x - 2y) - 4) + 4`

Explanation:

Given

`x^2 - 4xy - 4x + 4y^2 + 8y + 4`

Required

Factorize

`x^2 - 4xy - 4x + 4y^2 + 8y + 4`

Rearrange

`x^2 - 4xy + 4y^2- 4x + 8y + 4`

Group into three

`(x^2 - 4xy + 4y^2)- (4x - 8y) + (4)`

Factorize the first bracket

`(x^2 - 2xy- 2xy + 4y^2) - (4x - 8y) + (4)`

`x(x - 2y) - 2y(x - 2y)- (4x - 8y) + (4)`

`(x - 2y)(x - 2y) - (4x - 8y) + (4)`

Factorize the second bracket

`(x - 2y)(x - 2y) - 4(x - 2y) + 4`

x - 2y is a common factor of
`(x - 2y)(x - 2y) - 4(x - 2y)`

So, the expression becomes

`(x - 2y)((x - 2y) - 4) + 4`

The expression cannot be further simplified;

Hence,
`x^2 - 4xy - 4x + 4y^2 + 8y + 4` is equivalent to
`(x - 2y)((x - 2y) - 4) + 4`