Final answer:
The conic section represented by the equation x^2 + 4xy - 4y^2 - 2x + 4y + 5 = 0 is identified as a hyperbola because the discriminant (B^2 - 4AC) is positive.
Step-by-step explanation:
To identify the conic given by the equation x^2 + 4xy - 4y^2 - 2x + 4y + 5 = 0, we should first put it into a standard form to determine which type of conic section it represents. We do this by completing the square for both x and y terms. However, in this case, the presence of the mixed xy term suggests that this is not a parabola, circle or ellipse, because those do not have mixed xy terms in their general equations.
The general equation of a conic is A*x^2 + B*xy + C*y^2 + D*x + E*y + F = 0, where the classification depends on the values of A, B, and C. One way to classify the conic is by calculating the discriminant, which is given by B^2 - 4AC. If the discriminant is positive, the conic section is a hyperbola; if it is zero, the conic section is a parabola, and if it's negative, the conic is an ellipse or a circle.
For the given equation, A = 1, B = 4, and C = -4, so the discriminant is (4^2) - (4*1*-4) = 16 - (-16) = 32, which is positive. Thus, we can conclude that the conic section represented by the given equation is a hyperbola.