This problem is about finding the derivative of a rational function.
The function is f(x) = (2x² - 4x + 2) / (x² + 4x - 5). To find the derivative of f(x), we are going to apply the quotient rule, which states that for any two differentiable functions u and v, (u/v)' = (v * u' - u * v') / v².
Let's assign u = 2x² - 4x + 2 and v = x² + 4x - 5.
Next, we need to find u' and v'. Using the power rule that states (ax^n)' = a*n*x^(n-1):
u' = (2*2)x^(2-1) - (4*1)x^(1-1) + 0 = 4x - 4
and
v' = (1*2)x^(2-1) + 4*1*x^(1-1) - 0 = 2x + 4.
Now we can apply the quotient rule:
f'(x) = (v*u' - u*v') / v² = [(x² + 4x - 5)*(4x - 4) - (2x² - 4x + 2)*(2x + 4)] / (x² + 4x - 5)².
After simplifying the equation, we get the derivative of the function:
f'(x) = (-2x - 4)*(2x² - 4x + 2) / (x² + 4x - 5)² + (4x - 4) / (x² + 4x - 5).
This is the derivative of the given function.