Expanding and Simplifying the Equation: (xy+1)(x^2y^2xy+1)+(x^31)(1y^3)=x^3+y^3
This equation appears complex, but it can be simplified through algebraic manipulation. Let's break down the process step by step:
1. Expanding the Products
First, we need to expand the products on the left side of the equation using the distributive property (also known as FOIL method):

(xy + 1)(x²y²  xy + 1):
 xy * x²y² = x³y³
 xy * (xy) = x²y²
 xy * 1 = xy
 1 * x²y² = x²y²
 1 * (xy) = xy
 1 * 1 = 1

(x³  1)(1  y³):
 x³ * 1 = x³
 x³ * (y³) = x³y³
 1 * 1 = 1
 1 * (y³) = y³
2. Combining Like Terms
Now, let's combine the terms we obtained from expanding the products:
x³y³  x²y² + xy + x²y²  xy + 1 + x³  x³y³  1 + y³ = x³ + y³
Notice that several terms cancel out:
 x³y³ and x³y³ cancel
 x²y² and x²y² cancel
 xy and xy cancel
 1 and 1 cancel
This leaves us with:
x³ + y³ = x³ + y³
3. Conclusion
The simplified equation is x³ + y³ = x³ + y³. This is a true statement for any values of x and y. Therefore, the original equation is an identity, meaning it holds true for all values of its variables.