Step 1 :We are given that the polynomial $R(x)$ has degree 8 and has the following zeros: 2, -4, 5, and -5+3i.
Step 2 :From the conjugate zeros theorem, we know that if a polynomial has real coefficients and a complex number a + bi is a zero, then its conjugate a - bi is also a zero. Therefore, the conjugate of -5+3i, which is -5-3i, is also a zero of $R(x)$.
Step 3 :Thus, another zero of $R(x)$ is \(-5-3i\).
Step 4 :A polynomial of degree n can have at most n zeros. Since the degree of $R(x)$ is 8, it can have at most 8 zeros.
Step 5 :Since we already know 5 zeros (2, -4, 5, -5+3i, -5-3i), the maximum number of real zeros that $R(x)$ can have is 5.
Step 6 :Complex zeros always come in conjugate pairs, so the maximum number of nonreal zeros that $R(x)$ can have is the degree of the polynomial divided by 2, which is 4. However, since we already have a pair of nonreal zeros (-5+3i, -5-3i), the maximum number of nonreal zeros that $R(x)$ can have is 3.
Step 7 :Final Answer: (a) Another zero of $R(x)$ is \(\boxed{-5-3i}\). (b) The maximum number of real zeros that $R(x)$ can have is \(\boxed{5}\). (c) The maximum number of nonreal zeros that $R(x)$ can have is \(\boxed{3}\).