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

$$(2+kx{)}^6$$

where $k$ is a constant. Give each term in simplest form.

The population $P$ of a bacterial culture is modeled for small time intervals $x$ by the expression

$$(5-2.5x)(2+kx{)}^6$$

where $k$ is a growth constant. Given that the coefficient of $x^2$ in the expansion of this expression is $60$, find the possible values of $k$.