Use Vieta's formulas instead of solving the quadratic. For x^2 - (-9)x + (-10) = 0, the roots α and β satisfy α + β = -9 and αβ = -10. 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 = (-9)^2 - 2(-10) = 81 - (-20) = 101. This is why the correct answer is 101.