Divide exactly and identify real and imaginary parts

“Realising” the denominator means multiplying numerator and denominator by the denominator’s conjugate, here $2-3i$:

$\begin{aligned}z&=\frac{5-3i}{2+3i}\cdot\frac{2-3i}{2-3i}\\&=\frac{10-15i-6i+9i^2}{4+9}\\&=\frac{10-9-21i}{4+9}\\&=\frac1{13}-\frac{21}{13}i.\end{aligned}$

The denominator is real because $ww^*=x^2+y^2$ for $w=x+iy$: here it is $2^2+3^2=13$. Split the fraction to read off $\operatorname{Re}(z)=1/13$ and $\operatorname{Im}(z)=-21/13$. The imaginary part is the coefficient, without $i$.