Solving the Equation (x²  7)² + 2x²  14 = 0
This equation looks complex at first glance, but we can solve it by employing a few algebraic tricks. Here's how:
1. Simplifying the Equation

Recognize the pattern: Notice that the expression (x²  7)² is a perfect square. We can rewrite the equation as: (x²  7)² + 2(x²  7) = 0

Substitution: Let's make a substitution to simplify the equation further. Let y = (x²  7). Now the equation becomes: y² + 2y = 0
2. Solving the Quadratic Equation

Factoring: We can factor out a 'y' from the equation: y(y + 2) = 0

Zero Product Property: For the product of two factors to be zero, at least one of them must be zero. Therefore, we have two possible solutions:
 y = 0
 y + 2 = 0
3. Substituting Back and Solving for x

Substitute y back: Now we need to substitute back (x²  7) for y in both equations:
 x²  7 = 0
 x²  7 + 2 = 0

Solve for x:
 Equation 1: x² = 7 => x = ±√7
 Equation 2: x² = 5 => x = ±√5
4. Final Solution
The solutions to the equation (x²  7)² + 2x²  14 = 0 are:
x = √7, x = √7, x = √5, x = √5