The function $\text{f}$ is defined by $\text{f}(z) = z^4 + Az^3 + Bz^2 + Cz + 45$, where $A$, $B$ and $C$ are real numbers. Given that $2 + \text{i}$ is a root of $\text{f}(z) = 0$ and $(z - k)^2$ is a factor of $\text{f}(z)$, where $k$ is a positive real number, find the values of $A$, $B$, $C$ and $k$.

1. Identify the second complex root using the conjugate roots theorem.

Since the coefficients $A$, $B$, and $C$ of the polynomial $\text{f}(z)$ are real numbers, the complex roots must occur in conjugate pairs.
Therefore, $2 - \text{i}$ is also a root of $\text{f}(z) = 0$.

2. Form the quadratic factor corresponding to the complex conjugate roots.

Sum of complex roots $= (2 + \text{i}) + (2 - \text{i}) = 4$
Product of complex roots $= (2 + \text{i})(2 - \text{i}) = 2^2 - \text{i}^2 = 4 - (-1) = 5$

The corresponding quadratic factor is given by $z^2 - (\text{sum of roots})z + (\text{product of roots})$, which is:
$z^2 - 4z + 5$

3. Express the polynomial as a product of its quadratic factors and compare constant terms to find $k$.

Given that $\text{f}(z)$ is a polynomial of degree 4 with a leading coefficient of $1$, and $(z - k)^2$ is a factor, we can express $\text{f}(z)$ as:
$\text{f}(z) = (z^2 - 4z + 5)(z - k)^2$
$\text{f}(z) = (z^2 - 4z + 5)(z^2 - 2kz + k^2)$

Comparing the constant terms of both sides of the identity:
$5k^2 = 45$
$k^2 = 9$
Since $k$ is given as a positive real number, $k = 3$.

4. Expand the factorised form to find the values of $A$, $B$, and $C$.

Substitute $k = 3$ back into the expression:
$\text{f}(z) = (z^2 - 4z + 5)(z^2 - 6z + 9)$

Expanding the expression:
$\text{f}(z) = z^2(z^2 - 6z + 9) - 4z(z^2 - 6z + 9) + 5(z^2 - 6z + 9)$
$\text{f}(z) = z^4 - 6z^3 + 9z^2 - 4z^3 + 24z^2 - 36z + 5z^2 - 30z + 45$
$\text{f}(z) = z^4 - 10z^3 + 38z^2 - 66z + 45$

Comparing the coefficients with the original equation $\text{f}(z) = z^4 + Az^3 + Bz^2 + Cz + 45$:
$A = -10$
$B = 38$
$C = -66$