(a) To express \(\frac{2 + 3i}{1 - 2i}\) in the form \(re^{i\theta}\), multiply numerator and denominator by the conjugate of the denominator:

\(\frac{2 + 3i}{1 - 2i} \times \frac{1 + 2i}{1 + 2i} = \frac{(2 + 3i)(1 + 2i)}{(1 - 2i)(1 + 2i)}\)

Calculate the denominator: \((1 - 2i)(1 + 2i) = 1 + 2i - 2i - 4i^2 = 1 + 4 = 5\).

Calculate the numerator: \((2 + 3i)(1 + 2i) = 2 + 4i + 3i + 6i^2 = 2 + 7i - 6 = -4 + 7i\).

Thus, \(\frac{2 + 3i}{1 - 2i} = \frac{-4 + 7i}{5} = -\frac{4}{5} + \frac{7}{5}i\).

The modulus \(r\) is \(\sqrt{\left(-\frac{4}{5}\right)^2 + \left(\frac{7}{5}\right)^2} = \sqrt{\frac{16}{25} + \frac{49}{25}} = \sqrt{\frac{65}{25}} = \frac{\sqrt{65}}{5} \approx 1.61\).

The argument \(\theta\) is \(\arctan\left(\frac{7/5}{-4/5}\right) = \arctan\left(-\frac{7}{4}\right) \approx 2.09\).

(b) The equation \(|z - 3 + 2i| = 1\) represents a circle on the Argand diagram with center \((3, -2)\) and radius 1.

The least value of \(|z|\) is the distance from the origin to the closest point on the circle. The distance from the origin to the center \((3, -2)\) is \(\sqrt{3^2 + (-2)^2} = \sqrt{9 + 4} = \sqrt{13}\).

The least value of \(|z|\) is \(\sqrt{13} - 1\).