r4 is a variable that was introduced to make the solution more compact.
Notice under the first summation it says "r4 in #1", and then it gives a definition for #1 in terms of RootOf(). What that is telling you is that the RootOf() has multiple roots (four of them in this case) and that a list of those roots is to be calculated and that the summation should then proceed with r4 being substituted with each member of that list of roots.
The RootOf() is being applied to a quartic equation. There are closed form solutions for that quartic, but they are typically quite long to write out, and if you were to write it all out then the result would be very difficult to make sense of as a human.
So in this case, the inverse laplace has a definite value, and it would be possible to list it all out, but your mind would probably go on strike if you tried to understand it in the expanded form.
As I look at those coefficients in the image, it looks to me as if you attempted to do an symbolic inverse laplace on a system that involved floating point values. Symbolic inverse laplace attempts to find a exact closed form solution if possible, but finding exact solutions does not make sense when the original coefficients are uncertain.
For example, something "simple" like 1.1E-6 might look like an exact coefficient, but internally it is 1.10000000000000005610204141526242693771564518101513385772705078125e-06 and from a scientific point of view, writing 1.1E-6 represents "some value, X, such that 1.05E-6 <= X < 1.15E-6"
If possible, you should convert all of your coefficients to symbolic as they are introduced, so that you can reduce round-off error. For example, instead of 1.1E-6 you might use sym(11)/sym(10^7)
