Complete the Polynomial Equation: Solve for x and y Blanks in (x^2+y)^2+(y^2+x)^2

Fill in the blanks:

$(x^2+y)^2+(y^2+x)^2=?(x^2+1)+2xy\lbrack?\rbrack+y^2\lbrack?\rbrack$

To solve this problem, we'll follow these steps:

• Step 1: Expand the squares $(x^2+y)^2$ and $(y^2+x)^2$.
• Step 2: Combine like terms from the expansions.
• Step 3: Compare the result to the right-hand side of the equation.

Now, let's work through each step:

Step 1: Expand the expressions:
$(x^2 + y)^2 = (x^2)^2 + 2(x^2)(y) + y^2 = x^4 + 2x^2y + y^2$
$(y^2 + x)^2 = (y^2)^2 + 2(y^2)(x) + x^2 = y^4 + 2y^2x + x^2$

Step 2: Combine the expansions:
$(x^2+y)^2 + (y^2+x)^2 = x^4 + 2x^2y + y^2 + y^4 + 2y^2x + x^2$
Combine like terms:
$= x^4 + x^2 + y^4 + y^2 + 2xy(x+y)$

Step 3: Compare with $(x^2+1) + 2xy\lbrack?\rbrack + y^2\lbrack?\rbrack$:
The complete expression terms include $(x^2+1)$, a term linked to $2xy(x+y)$, and a term in $y^2$ which should account for $y^4 + y^2$ similarity.

Therefore, the solution to the problem is that the blanks should be filled with $x^2,(x+y),y^2+1$.