Question

The factor of 2ax^2+4axy+3bx^2+2ay^2+6bxy^+3by^2

Answer 1

`2ax^2+4axy+3bx^2+2ay^2+6bxy+3by^2= (x+y)^2(2a+3b)`

Step-by-step explanation:

Given

`2ax^2+4axy+3bx^2+2ay^2+6bxy+3by^2`

Required

Factorize

`2ax^2+4axy+3bx^2+2ay^2+6bxy+3by^2`

Rewrite as:

`2ax^2+4axy+2ay^2+3bx^2+6bxy+3by^2`

Use square parenthesis

`[2ax^2+4axy+2ay^2]+[3bx^2+6bxy+3by^2]`

Expand each bracket

`[2ax^2+2axy+2axy+2ay^2]+[3bx^2+3bxy+3bxy+3by^2]`

Factorize each:

`[2ax(x+y) + 2ay(x+y)]+[3bx(x+y)+3by(x+y)]`

`[(2ax+2ay)(x+y)]+[(3bx+3by)(x+y)]`

Further, factorize:

`(x+y)[(2ax+2ay)]+[(3bx+3by)]`

`(x+y)[(2ax+2ay)+(3bx+3by)]`

Remove bracket

`(x+y)[2ax+2ay+3bx+3by]`

Reorder:

`(x+y)[2ax+3bx+2ay+3by]`

Factorize:

`(x+y)[x(2a+3b)+y(2a+3b)]`

`(x+y)[(x+y)(2a+3b)]`

Remove square bracket

`(x+y)(x+y)(2a+3b)`

This gives:

`(x+y)^2(2a+3b)`

Hence:

`2ax^2+4axy+3bx^2+2ay^2+6bxy+3by^2= (x+y)^2(2a+3b)`