Complexification of (e^(-5t))cos(3t)

Given the second order equation: y''+4y'+20y=(e^(-5t))cos(3t)

Homogenous solution is k1e^(-2t)cos(4t) + k2e^(-2t)sin(4t) I'm pretty sure, but I'm having trouble complexifying the right hand side. Please help

Complex analysis is not my forte, but could the fact that:

Re(e^((-5 + 3i)t))
= Re(e^(5t) * e^(3it))
= Re(e^(-5t)(cos(3t) + i sin(3t))
= e^(-5t)cos(3t)

help in any way?