Find the first 4 terms, in ascending powers of $x$, of the binomial expansion of

$${\left(3+\frac{kx}{9}\right)}^6$$

where $k$ is a non-zero constant. Give each term in its simplest form.

Given that

$$f(x)=(4+2x){\left(3+\frac{kx}{9}\right)}^6$$

and that in the expansion of $f(x)$, the coefficient of $x$ is equal to the constant term, find the value of $k$.