So what you are going to do is assume that there is a function of the form
where is a function of x.
Now you will need to take three derivatives of this new function and plug them into the ODE.
I.e here is the first one
Dont forget to use the product rule.
You will end up with an ODE in u and make the substituion that Danny recommended and solve for what u needs to be.
Good luck.
The form of the given solution tells you that the charateristic equation has two complex roots and one real root.
This tells you that the two complex roots of the characteristic equation are -1+i, and -1-i (since the characteristic equation has real coefficients complex roots occur in conjugate pairs and only a linear combinatiation of the solutions corresponding to these roots will give the solution that you have been given).
This should be enough information to allow you to find the third root.
CB