Use Vieta's formulas instead of solving the quadratic. For x^2 - (-2)x + (-24) = 0, the roots α and β satisfy α + β = -2 and αβ = -24. The expression we need is α^2 + β^2. Use the identity α^2 + β^2 = (α + β)^2 - 2αβ. So α^2 + β^2 = (-2)^2 - 2(-24) = 4 - (-48) = 52. This is why the correct answer is 52.