Use Vieta's formulas instead of solving the quadratic. For x^2 - (-5)x + (-14) = 0, the roots α and β satisfy α + β = -5 and αβ = -14. 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 = (-5)^2 - 2(-14) = 25 - (-28) = 53. This is why the correct answer is 53.