Alright, so we have
[tex]2 (x^{2} + y^{2} )^{2} + x^{4}+y^{4})= [blank]+x^{4}+y^{4}[/tex] to fill in the first box. Since the general equation for expanding (x+y)² is x²+2xy+y², we can substitute x² for x and y² for y to get [tex](x^{2} + y^{2} )^{2}=x^{4}+2x^{2}y^{2}+y^{4}[/tex], making it
[tex] (x^{2} + y^{2} )^{2} + x^{4}+y^{4}=x^{4}+2x^{2}y^{2}+y^{4}+ x^{4}+y^{4} [/tex]
Crossing out one x^4 and one y^4 from both sides, we're left with the blank being
[tex](x^{2} + y^{2} )^{2}=x^{4}+2x^{2}y^{2}+y^{4}[/tex]
For the second blank, we can make x^4+x^4=2x^4 and y^4+y^4=2y^4, resulting in
[tex] (x^{2} + y^{2} )^{2} + x^{4}+y^{4}=x^{4}+2x^{2}y^{2}+y^{4}+ x^{4}+y^{4} =2x^{2}y^{2}+2x^{4}+2x^{4}[/tex]
Onto the last blank, we have [tex]2x^{2}y^{2}+2x^{4}+2x^{4}[/tex].
We can simply factor the two - this means that we divide the whole line by 2 and multiply the quotient by 2 to have it equal the original line - to get
[tex]2 (x^{2} + y^{2} )^{2} + x^{4}+y^{4})[/tex]
Feel free to ask further questions!
