Answer:
[tex]2ax^2+4axy+3bx^2+2ay^2+6bxy+3by^2= (x+y)^2(2a+3b)[/tex]
Explanation:
Given
[tex]2ax^2+4axy+3bx^2+2ay^2+6bxy+3by^2[/tex]
Required
Factorize
[tex]2ax^2+4axy+3bx^2+2ay^2+6bxy+3by^2[/tex]
Rewrite as:
[tex]2ax^2+4axy+2ay^2+3bx^2+6bxy+3by^2[/tex]
Use square parenthesis
[tex][2ax^2+4axy+2ay^2]+[3bx^2+6bxy+3by^2][/tex]
Expand each bracket
[tex][2ax^2+2axy+2axy+2ay^2]+[3bx^2+3bxy+3bxy+3by^2][/tex]
Factorize each:
[tex][2ax(x+y) + 2ay(x+y)]+[3bx(x+y)+3by(x+y)][/tex]
[tex][(2ax+2ay)(x+y)]+[(3bx+3by)(x+y)][/tex]
Further, factorize:
[tex](x+y)[(2ax+2ay)]+[(3bx+3by)][/tex]
[tex](x+y)[(2ax+2ay)+(3bx+3by)][/tex]
Remove bracket
[tex](x+y)[2ax+2ay+3bx+3by][/tex]
Reorder:
[tex](x+y)[2ax+3bx+2ay+3by][/tex]
Factorize:
[tex](x+y)[x(2a+3b)+y(2a+3b)][/tex]
[tex](x+y)[(x+y)(2a+3b)][/tex]
Remove square bracket
[tex](x+y)(x+y)(2a+3b)[/tex]
This gives:
[tex](x+y)^2(2a+3b)[/tex]
Hence:
[tex]2ax^2+4axy+3bx^2+2ay^2+6bxy+3by^2= (x+y)^2(2a+3b)[/tex]
