I am given a linear combination of wave function of HYDROGEN ATOM Ψ=1/2(Ψ200 +Ψ310+Ψ311+Ψ31-1), where the subscripts are n, l, m respectively.

I was asked to find all the possible outcomes when measuring Lx and their corresponding probabilities.

Should I construct matrices of the raising and lowering operator? How should I do so?
I think (I guess?) I can find the eigenfunction of Lx first by constructing the matrices of raising and lowering operator, by the relation of Lx=1/2(L+ + L_). Then I can find the eigenvalues, which are the possible outcome when I measure Lx.

I don't really know how to deal with a linear combination of wave functions.