 Second Chern class of a tensor product of rank-two bundles

The splitting principle for complex vector bundles makes pullback injective in integral cohomology and splits both bundles into lines with Chern roots $x_1,x_2$ and $y_1,y_2$. The tensor product has roots $x_i+y_j$. Expanding their six pairwise products gives the displayed formula. This integral computation does not divide by two, so it remains valid when the cohomology has two-torsion. In particular, taking $F$ trivial gives $c_2(E\oplus E)=c_1(E{)}^2+2c_2(E)$.