Answer
The charge of the particle is $~~-4~\mu C$
Work Step by Step
In part (a), we found that the charge on the inner surface of the shell is $~~+4~\mu C$
Since the electric field inside the conducting shell is zero, if we draw a Gaussian sphere with a surface within the shell that is concentric with the shell, then the enclosed charge must be zero.
Therefore, the charge of the particle must be $~~-4~\mu C$
