To divide a complex a+ib by a complex number c+id we have to multiply the numerator and denominator by the complex conjugate of the denominator of the original fraction; after that we make the operations.
With this in mind in this case we have:
[tex]\begin{gathered} \frac{1-4i}{3+2i}\cdot\frac{3-2i}{3-2i}=\frac{3-2i-12i+8i^2}{9-6i+6i-4i^2} \\ =\frac{3-14i-8}{9+4} \\ =\frac{-5-14i}{13} \\ =-\frac{5}{13}-\frac{14}{13}i \end{gathered}[/tex]Therefore the result of the division given is:
[tex]-\frac{5}{13}-\frac{14}{13}i[/tex]