Problem Statement:
Find the values of f(x²) and f(x⁴) given the equation 2f(x²) + 3f(1/x²) = x² - 1 for all x ∈ ℝ-{0}.

Solution:
To find the values of f(x²) and f(x⁴), we need to manipulate the given equation and solve for these unknowns.

Step 1: Manipulating the equation
Let's substitute x² with t in the equation to simplify the expressions.
2f(t) + 3f(1/t) = t - 1 (Equation 1)

Step 2: Finding f(1/t)
To find f(1/t), let's substitute t with 1/x² in Equation 1.
2f(1/x²) + 3f(x²) = 1/x² - 1 (Equation 2)

Step 3: Solving the system of equations
To solve the system of equations formed by Equation 1 and Equation 2, we can use substitution.

Step 3.1: Isolating f(t) in Equation 1
From Equation 1, we can isolate f(t) as follows:
2f(t) = t - 1 - 3f(1/t)
f(t) = (t - 1 - 3f(1/t))/2 (Equation 3)

Step 3.2: Substituting f(t) in Equation 2
Substituting Equation 3 into Equation 2, we get:
2f(1/x²) + 3((1/x²) - 1 - 3f(1/(1/x²)))/2) = 1/x² - 1

Simplifying this equation, we have:
2f(1/x²) + (3/x²) - 3 - 9f(x²) = 1/x² - 1

Step 3.3: Rearranging the equation
Rearranging the equation, we get:
9f(x²) - 2f(1/x²) = (3/x²) - 4

Step 4: Finding f(x²) and f(x⁴)
To find the values of f(x²) and f(x⁴), we need to solve the system of equations formed by Equation 1 and Equation 2.

Step 4.1: Substituting t with x² in Equation 3
Substituting t with x² in Equation 3, we get:
f(x²) = (x² - 1 - 3f(1/x²))/2

Step 4.2: Substituting Equation 3 into Equation 2
Substituting Equation 3 into Equation 2, we have:
9f(x²) - 2f(1/x²) = (3/x²) - 4

Step 4.3: Solving the system of equations
Now, we can substitute f(x²) from Step 4.1 into Equation 4.2 and solve for f(1/x