Twice at Five Units Out

A probe drifts along a straight path. Its position vector, measured from a base station at the origin, is

$\vec{p}(t) = \vec{a} + t\,\vec{b}, \qquad \vec{a} = (3,\ -1), \quad \vec{b} = (-2,\ 4)$

where the parameter $t$ may be any real number (negative values describe where the probe was before the log begins).

The probe is exactly $5$ units from the base station at more than one value of $t$.

What is the sum of all such values of $t$?