The situation becomes considerable harder beyond testing for a root of multiplicity $2$. The problem itself is very old. A recent article about this is "On equations defining coincident root loci" by J. Chipalkatti. The result is that the algebraic conditions can be expressed through a cohomology group of a certain complex of $SL_2$ representations.
Note: This is addressing the general question of when a polynomial has roots of multiplicities indexed by a partition $\lambda$. The question itself is about $\lambda=(3,1^{n-3})$.
The situation becomes considerable harder beyond testing for a root of multiplicity $2$. The problem itself is very old. A recent article about this is "On equations defining coincident root loci" by J. Chipalkatti. The result is that the algebraic conditions can be expressed through a cohomology group of a certain complex of $SL_2$ representations.