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This problem trains for: AMC-10, Math Kangaroo 9-10, SAT-II.

Find the sum of all the possible values of the parameter m for which the quadratic function with roots at:

has an axis of symmetry at:

The axis of symmetry of a parabola is located midway between the two roots. The midpoint of the roots has x-coordinate:

The requirement is to set this coordinate to be equal to 2:

which leads to a quadratic equation with variable m:

To find the sum of all the possible roots of this equation in the simplest manner, we apply Viète's relations: