If the roots of the equation x2 + 2ax + b = 0 are real and distinct and they differ by at most 2m, then b lies in the interval:
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(a2 - m2, a2)
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[a2 - m2, a2)
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[a2, a2 + m2]
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(a2, a2 + m2)
Let roots be alpha and beta. |alpha - beta| <= 2m. (alpha-beta)^2 <= 4m^2. (alpha+beta)^2 - 4alpha*beta <= 4m^2. (-2a)^2 - 4b <= 4m^2. 4a^2 - 4b <= 4m^2. a^2 - b <= m^2, so b >= a^2 - m^2. Since roots are real and distinct, discriminant > 0, so 4a^2 - 4b > 0, b < a^2. Thus b is in [a^2 - m^2, a^2).
Let the roots of the quadratic equation be alpha and beta. Using the relation between roots and coefficients, we have alpha + beta = -2a and alpha * beta = b. The difference between the roots is |alpha - beta| = sqrt((alpha + beta)2 - 4*alpha*beta). Substituting the coefficient values gives |alpha - beta| = sqrt(4a2 - 4b) = 2*sqrt(a2 - b). Because the roots are real and distinct, b must be less than a2, and the difference is at least zero, so a2 - b >= 0. The condition that the roots differ by at most 2m means 2*sqrt(a2 - b) <= 2m, which simplifies to a2 - b <= m2. Solving a2 - m2 <= b < a2 shows that b lies in the interval [a2 - m2, a2).