Solving the Equation (x^2 + 1)(x^2 + y^2) = 4x^2y
This equation presents an interesting challenge in algebraic manipulation and solution finding. Let's break down the process of solving it:
Expanding and Rearranging

Expand the left side: (x^2 + 1)(x^2 + y^2) = x^4 + x^2y^2 + x^2 + y^2

Rearrange the equation: x^4 + x^2y^2 + x^2 + y^2  4x^2y = 0
Recognizing a Pattern
The equation now resembles a quadratic form if we consider x^2 as a single variable. Let's substitute:
 Let 'a' = x^2
This gives us:
 a^2 + ay^2 + a + y^2  4ay = 0
Solving the Quadratic
Now we have a quadratic equation in terms of 'a'. We can solve for 'a' using the quadratic formula:
 a = [b ± √(b^2  4ac)] / 2a
Where:
 a = 1 (coefficient of a^2)
 b = y^2  4y + 1 (coefficient of a)
 c = y^2 (constant term)
Finding Solutions
 Solve for 'a' using the quadratic formula. This will give you two possible values for 'a'.
 Substitute back 'x^2' for 'a'. This will give you two equations in terms of x^2.
 Solve for 'x' in each equation. This might lead to real or complex solutions, depending on the value of 'y'.
 For each value of 'x', solve for 'y' using the original equation.
Note: The solutions might be complex numbers, depending on the values obtained during the solving process.
Conclusion
The equation (x^2 + 1)(x^2 + y^2) = 4x^2y can be solved by recognizing a quadratic form in terms of x^2, using the quadratic formula, and then solving for x and y individually. The solutions might involve complex numbers.