Solve what "using the Laplace transform" of that function?
If you mean just "find the Laplace transform of \(\displaystyle f(t)= cos(4t)e^{5t}\)" use the definition of the Laplace transform: \(\displaystyle \int_0^\infty cos(4t)e^{5t}e^{-st}dt= \int_0^\infty cos(4t)e^{(5-s)t}dt\)
That integral can be done using "integration by parts"- twice.
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