Use Vieta's formulas instead of solving the quadratic. For x^2 - (4)x + (-21) = 0, the roots α and β satisfy α + β = 4 and αβ = -21. 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 = (4)^2 - 2(-21) = 16 - (-42) = 58. This is why the correct answer is 58.