Use Vieta's formulas instead of solving the quadratic. For x^2 - (-7)x + (-8) = 0, the roots α and β satisfy α + β = -7 and αβ = -8. 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 = (-7)^2 - 2(-8) = 49 - (-16) = 65. This is why the correct answer is 65.