Solving the Equation (x^4 + x^2 + 1)(x^4 + x^2 + 2) = 12
This problem involves solving a rather complex equation with highdegree terms. We can solve it by simplifying the equation and then using various algebraic techniques.
Simplifying the Equation

Substitution: Let's make the equation easier to work with by introducing a substitution. Let y = x^2. This simplifies the equation to:
(y^2 + y + 1)(y^2 + y + 2) = 12

Expansion: Expand the left side of the equation:
y^4 + 2y^3 + 4y^2 + 3y + 2 = 12

Rearranging: Move the constant term to the left side:
y^4 + 2y^3 + 4y^2 + 3y  10 = 0
Solving the Quartic Equation
Now we have a quartic equation (an equation with the highest power of the variable being 4). Solving quartic equations can be quite complex, but we can try to factor it:

Factoring: We can factor the equation as follows:
(y^2 + 2y  2)(y^2 + 5) = 0

Solving the Factors: Now we have two quadratic equations to solve:

y^2 + 2y  2 = 0 This equation can be solved using the quadratic formula:
y = [2 ± √(2^2  4 * 1 * 2)] / (2 * 1) y = [2 ± √12] / 2 y = 1 ± √3

y^2 + 5 = 0 This equation has no real solutions, as the square of a real number cannot be negative.

Finding the Solutions for x
Remember that y = x^2. We need to substitute back and solve for x:

For y = 1 + √3:
x^2 = 1 + √3 x = ±√(1 + √3)

For y = 1  √3:
x^2 = 1  √3 x = ±√(1  √3)
The solutions for x involve complex numbers due to the square root of negative values.
Therefore, the equation (x^4 + x^2 + 1)(x^4 + x^2 + 2) = 12 has four solutions, all of which are complex numbers.