$\displaystyle \frac{4+3j}{200}(\frac{-3+4j}{25})^n+\frac{4-3j}{200}(\frac{-3-4j}{25})^n$

I know 100% that this expression must be real since the excitation to the system was real in my Linear, time-invariant system of differential equations.

Somehow, the solutions manual transforms the equation into some e^(jw)s, and uses Euler's identity from there to reach two real sinusoidal functions. The problem I'm having is making the e^(jw) out of that expression in the first place.